7.8 IMPROPER INTEGRALS

Size: px
Start display at page:

Download "7.8 IMPROPER INTEGRALS"

Transcription

1 7.8 Improper Integrls 547 the grph of g psses through the points (, ), (, ), nd (, ); the grph of g psses through the points (, ), ( 3, 3 ), nd ( 4, 4 );... the grph of g n/ psses through the points ( n, n ), ( n, n ), nd ( n, n ). Verif tht Formul (8) computes the re under piecewise qudrtic function showing tht ( n/ ) j g j () j= j = ( ) n + n + 4 n + n ] 55. Writing Discuss two different circumstnces under which numericl integrtion is necessr. 56. Writing For the numericl integrtion methods of this section, etter ccurc of n pproimtion ws otined incresing the numer of sudivisions of the intervl. Another strteg is to use the sme numer of suintervls, ut to select suintervls of differing lengths. Discuss scheme for doing this to pproimte 4 using trpezoidl pproimtion with 4 suintervls. Comment on the dvntges nd disdvntges of our scheme. QUICK CHECK ANSWERS 7.7 (. () (L n + R n ) ( n 8 ) n + n ]. M n <I<T n 3. () 3 M T 3 ) ( ) 4. () 5. () M = T = 9 (c) S = (c),8, 7.8 IMPROPER INTEGRALS Up to now we hve focused on definite integrls with continuous integrnds nd finite intervls of integrtion. In this section we will etend the concept of definite integrl to include infinite intervls of integrtion nd integrnds tht ecome infinite within the intervl of integrtion. IMPROPER INTEGRALS It is ssumed in the definition of the definite integrl f() tht,] is finite intervl nd tht the limit tht defines the integrl eists; tht is, the function f is integrle. We oserved in Theorems 5.5. nd tht continuous functions re integrle, s re ounded functions with finitel mn points of discontinuit. We lso oserved in Theorem tht functions tht re not ounded on the intervl of integrtion re not integrle. Thus, for emple, function with verticl smptote within the intervl of integrtion would not e integrle. Our min ojective in this section is to etend the concept of definite integrl to llow for infinite intervls of integrtion nd integrnds with verticl smptotes within the intervl of integrtion. We will cll the verticl smptotes infinite discontinuities, nd we will cll

2 548 Chpter 7 / Principles of Integrl Evlution integrls with infinite intervls of integrtion or infinite discontinuities within the intervl of integrtion improper integrls. Here re some emples: Improper integrls with infinite intervls of integrtion:, e, + Improper integrls with infinite discontinuities in the intervl of integrtion: 3 3,, tn Improper integrls with infinite discontinuities nd infinite intervls of integrtion:, 9, π sec Figure 7.8. = Figure 7.8. Are = + f() = INTEGRALS OVER INFINITE INTERVALS To motivte resonle definition for improper integrls of the form f() let us egin with the cse where f is continuous nd nonnegtive on,+ ), so we cn think of the integrl s the re under the curve = f()over the intervl,+ ) (Figure 7.8.). At first, ou might e inclined to rgue tht this re is infinite ecuse the region hs infinite etent. However, such n rgument would e sed on vgue intuition rther thn precise mthemticl logic, since the concept of re hs onl een defined over intervls of finite etent. Thus, efore we cn mke n resonle sttements out the re of the region in Figure 7.8., we need to egin defining wht we men the re of this region. For tht purpose, it will help to focus on specific emple. Suppose we re interested in the re A of the region tht lies elow the curve = / nd ove the intervl, + ) on the -is. Insted of tring to find the entire re t once, let us egin clculting the portion of the re tht lies ove finite intervl,], where >is ritrr. Tht re is ] = = (Figure 7.8.). If we now llow to increse so tht +, then the portion of the re over the intervl,] will egin to fill out the re over the entire intervl, + ) (Figure 7.8.3), nd hence we cn resonl define the re A under = / over the intervl, + ) to e ( A = = lim + = lim ) = () + Thus, the re hs finite vlue of nd is not infinite s we first conjectured. = = = = Figure Are = 3 Are = 3 Are = Are = With the preceding discussion s our guide, we mke the following definition (which is pplicle to functions with oth positive nd negtive vlues).

3 7.8 Improper Integrls 549 If f is nonnegtive over the intervl,+ ), then the improper integrl in Definition 7.8. cn e interpreted to e the re under the grph of f over the intervl,+ ). If the integrl converges, then the re is finite nd equl to the vlue of the integrl, nd if the integrl diverges, then the re is regrded to e infinite definition The improper integrl of f over the intervl,+ ) is defined to e f() = lim f() + In the cse where the limit eists, the improper integrl is sid to converge, nd the limit is defined to e the vlue of the integrl. In the cse where the limit does not eist, the improper integrl is sid to diverge, nd it is not ssigned vlue. Emple Evlute () 3 Solution (). Following the definition, we replce the infinite upper limit finite upper limit, nd then tke the limit of the resulting integrl. This ields = lim 3 + = lim ] ( = lim ) = Since the limit is finite, the integrl converges nd its vlue is /. = 3 = = Figure Solution. = lim + = lim ] ln + = lim ln =+ + In this cse the integrl diverges nd hence hs no vlue. Becuse the functions / 3, /, nd / re nonnegtive over the intervl, + ), it follows from () nd the lst emple tht over this intervl the re under = / 3 is, the re under = / is, nd the re under = / is infinite. However, on the surfce the grphs of the three functions seem ver much like (Figure 7.8.4), nd there is nothing to suggest wh one of the res should e infinite nd the other two finite. One eplntion is tht / 3 nd / pproch zero more rpidl thn / s +, so tht the re over the intervl,] ccumultes less rpidl under the curves = / 3 nd = / thn under = / s +, nd the difference is just enough tht the first two res re finite nd the third is infinite. Emple For wht vlues of p does the integrl converge? p Solution. We know from the preceding emple tht the integrl diverges if p =, so let us ssume tht p =. In this cse we hve = lim p p ] p = lim = lim p + + p + p ] p If p>, then the eponent p is negtive nd p s + ; nd if p<, then the eponent p is positive nd p + s +. Thus, the integrl converges if p>nd diverges otherwise. In the convergent cse the vlue of the integrl is p = p ] = p (p > )

4 55 Chpter 7 / Principles of Integrl Evlution The following theorem summrizes this result theorem p = if p> p diverges if p Emple 3 Evlute ( )e. Solution. We egin evluting the indefinite integrl using integrtion prts. Setting u = nd dv = e ields ( )e = e ( ) e = e + e + e + C = e + C = ( )e 3 The net signed re etween the grph nd the intervl, + ) is zero. Figure If f is nonnegtive over the intervl (, + ), then the improper integrl f() cn e interpreted to e the re under the grph of f over the intervl (, + ). The re is finite nd equl to the vlue of the integrl if the integrl converges nd is infinite if it diverges. Thus, ( )e = lim + ( )e = lim e ] = lim + + e The limit is n indeterminte form of tpe /, so we will ppl L Hôpitl s rule differentiting the numertor nd denomintor with respect to. This ields ( )e = lim + e = We cn interpret this to men tht the net signed re etween the grph of = ( )e nd the intervl, + ) is (Figure 7.8.5) definition The improper integrl of f over the intervl (,] is defined to e f() = f() () lim The integrl is sid to converge if the limit eists nd diverge if it does not. The improper integrl of f over the intervl (, + ) is defined s f() = c f() + c f() (3) where c is n rel numer. The improper integrl is sid to converge if oth terms converge nd diverge if either term diverges. Emple 4 Evlute +. Although we usull choose c = in (3), the choice does not mtter ecuse it cn e proved tht neither the convergence nor the vlue of the integrl is ffected the choice of c. Solution. we otin + = lim + We will evlute the integrl choosing c = in (3). With this vlue for c + = lim ] + = lim tn = lim + + (tn ) = π + = lim tn ] = lim ( tn ) = π

5 7.8 Improper Integrls 55 Thus, the integrl converges nd its vlue is + = + = = π + π Are = c + = π Since the integrnd is nonnegtive on the intervl (, + ), the integrl represents the Figure re of the region shown in Figure f () INTEGRALS WHOSE INTEGRANDS HAVE INFINITE DISCONTINUITIES Net we will consider improper integrls whose integrnds hve infinite discontinuities. We will strt with the cse where the intervl of integrtion is finite intervl,] nd the infinite discontinuit occurs t the right-hnd endpoint. To motivte n pproprite definition for such n integrl let us consider the cse where f is nonnegtive on,], so we cn interpret the improper integrl f() s the re of the region in Figure The prolem of finding the re of this region is complicted the fct tht it etends indefinitel in the positive -direction. However, insted of tring to find the entire re t once, we cn proceed indirectl clculting the portion of the re over the intervl,k], where k<, nd then letting k pproch to fill out the re of the entire region (Figure 7.8.7). Motivted this ide, we mke the following definition. () definition If f is continuous on the intervl,], ecept for n infinite discontinuit t, then the improper integrl of f over the intervl, ] is defined s k f () f() = lim k k f() (4) k Figure = Figure In the cse where the indicted limit eists, the improper integrl is sid to converge, nd the limit is defined to e the vlue of the integrl. In the cse where the limit does not eist, the improper integrl is sid to diverge, nd it is not ssigned vlue. Emple 5 Evlute. Solution. The integrl is improper ecuse the integrnd pproches + s pproches the upper limit from the left (Figure 7.8.8). From (4), k = lim k = lim = lim k + k k ] = ] k Improper integrls with n infinite discontinuit t the left-hnd endpoint or inside the intervl of integrtion re defined s follows.

6 55 Chpter 7 / Principles of Integrl Evlution definition If f is continuous on the intervl,], ecept for n infinite discontinuit t, then the improper integrl of f over the intervl, ] is defined s f() = lim k + k f() (5) The integrl is sid to converge if the indicted limit eists nd diverge if it does not. If f is continuous on the intervl,], ecept for n infinite discontinuit t point c in (, ), then the improper integrl of f over the intervl, ] is defined s c f () f() c c f() = c f() + c f() (6) where the two integrls on the right side re themselves improper. The improper integrl on the left side is sid to converge if oth terms on the right side converge nd diverge if either term on the right side diverges (Figure 7.8.9). f () is improper. Emple 6 Evlute Figure () 4 ( ) /3 Solution (). The integrl is improper ecuse the integrnd pproches s pproches the lower limit from the right (Figure 7.8.). From Definition we otin = lim k + k = lim k + ] ln k = lim k + ] ln +ln k = lim ln k = k + = so the integrl diverges. Solution. The integrl is improper ecuse the integrnd pproches + t =, which is inside the intervl of integrtion. From Definition we otin Figure ( ) /3 = ( ) /3 + 4 ( ) /3 (7) nd we must investigte the convergence of oth improper integrls on the right. Since 4 = lim ( ) /3 k = lim ( ) /3 k + k 4 k = lim 3(k ) /3 3( ) /3] = 3 ( ) /3 k = lim 3(4 ) /3 3(k ) /3] = 3 3 ( ) /3 k + we hve from (7) tht 4 ( ) /3 =

7 7.8 Improper Integrls 553 WARNING It is sometimes tempting to ppl the Fundmentl Theorem of Clculus directl to n improper integrl without tking the pproprite limits. To illustrte wht cn go wrong with this procedure, suppose we fil to recognize tht the integrl (8) ( ) is improper nd mistkenl evlute this integrl s ] = () = This result is clerl incorrect ecuse the integrnd is never negtive nd hence the integrl cnnot e negtive! To evlute (8) correctl we should first write ( ) = ( ) + nd then tret ech term s n improper integrl. For the first term, so (8) diverges. ( ) = lim k k ( ) = lim k ( ) k ] =+ ARC LENGTH AND SURFACE AREA USING IMPROPER INTEGRALS In Definitions 6.4. nd 6.5. for rc length nd surfce re we required the function f to e smooth (continuous first derivtive) to ensure the integrilit in the resulting formul. However, smoothness is overl restrictive since some of the most sic formuls in geometr involve functions tht re not smooth ut led to convergent improper integrls. Accordingl, let us gree to etend the definitions of rc length nd surfce re to llow functions tht re not smooth, ut for which the resulting integrl in the formul converges. Emple 7 Derive the formul for the circumference of circle of rdius r. r Figure 7.8. = r r Solution. For convenience, let us ssume tht the circle is centered t the origin, in which cse its eqution is + = r. We will find the rc length of the portion of the circle tht lies in the first qudrnt nd then multipl 4 to otin the totl circumference (Figure 7.8.). Since the eqution of the upper semicircle is = r, it follows from Formul (4) of Section 6.4 tht the circumference C is r r ( ) C = 4 + (d/) = 4 + r r = 4r r This integrl is improper ecuse of the infinite discontinuit t = r, nd hence we evlute it writing C = 4r lim k r k k r = 4r lim = 4r lim k r r ( sin )] k r ( k sin r ) sin Formul (77) in the Endpper Integrl Tle ( π ) = 4rsin sin ]=4r = πr ]

8 554 Chpter 7 / Principles of Integrl Evlution QUICK CHECK EXERCISES 7.8 (See pge 557 for nswers.). In ech prt, determine whether the integrl is improper, nd if so, eplin wh. Do not evlute the integrls. () (c) 3π/4 π/4 cot + (d) π π/4 cot. Epress ech improper integrl in Quick Check Eercise in terms of one or more pproprite limits. Do not evlute the limits. 3. The improper integrl p converges to provided. 4. Evlute the integrls tht converge. () (c) e 3 (d) e 3 EXERCISE SET 7.8 Grphing Utilit C CAS. In ech prt, determine whether the integrl is improper, nd if so, eplin wh. 5 5 () (c) ln (d) e (e) 3 (f ) π/4 tn. In ech prt, determine ll vlues of p for which the integrl is improper. () (c) e p p p 3 3 Evlute the integrls tht converge e 4 π/ e ln 3 8. ( ) 3. e ( + 3) 8. ( 4) tn e ln + 9 e 3 e + e t dt + e t π/ π/3 3 8 sin cos / ( + ) 3. π/4 sec tn ( ) /3 ( + ) True Flse Determine whether the sttement is true or flse. Eplin our nswer /3 converges to If f is continuous on,+ ] nd lim + f() =, then + f() converges. 35. is n improper integrl. ( 3) = 37 4 Mke the u-sustitution nd evlute the resulting definite integrl. e 37. ; u = Note: u + s +.] ( + 4) ; u = e ; u = e e Note: u s +.] Note: u + s +.]

9 C C C 4. e ; u = e e 4 4 Epress the improper integrl s limit, nd then evlute tht limit with CAS. Confirm the nswer evluting the integrl directl with the CAS e cos 4. e In ech prt, tr to evlute the integrl ectl with CAS. If our result is not simple numericl nswer, then use the CAS to find numericl pproimtion of the integrl. () (c) ln e (d) + 3 sin In ech prt, confirm the result with CAS. sin π + () = (c) ln = π + e = π 45. Find the length of the curve = (4 /3 ) 3/ over the intervl, 8]. 46. Find the length of the curve = 4 over the intervl, ] Use L Hôpitl s rule to help evlute the improper integrl. ln 47. ln Find the re of the region etween the -is nd the curve = e 3 for. 5. Find the re of the region etween the -is nd the curve = 8/( 4) for Suppose tht the region etween the -is nd the curve = e for is revolved out the -is. () Find the volume of the solid tht is generted. Find the surfce re of the solid Use the results in Eercise Improper Integrls () Confirm grphicll nd lgericll tht e e ( ) Evlute the integrl e (c) Wht does the result otined in prt tell ou out the integrl e? 54. () Confirm grphicll nd lgericll tht + e ( ) + Evlute the integrl + (c) Wht does the result otined in prt tell ou out the integrl e +? 55. Let R e the region to the right of = tht is ounded the -is nd the curve = /. When this region is revolved out the -is it genertes solid whose surfce is known s Griel s Horn (for resons tht should e cler from the ccompning figure). Show tht the solid hs finite volume ut its surfce hs n infinite re. Note: It hs een suggested tht if one could sturte the interior of the solid with pint nd llow it to seep through to the surfce, then one could pint n infinite surfce with finite mount of pint! Wht do ou think?] = FOCUS ON CONCEPTS 5. Suppose tht f nd g re continuous functions nd tht f() g() if. Give resonle informl rgument using res to eplin wh the following results re true. () If f() diverges, then g() diverges. If g() converges, then f() converges nd f() g(). Note: The results in this eercise re sometimes clled comprison tests for improper integrls.] Figure E In ech prt, use Eercise 5 to determine whether the integrl converges or diverges. If it converges, then use prt of tht eercise to find n upper ound on the vlue of the integrl. () (c) e + 5 +

10 556 Chpter 7 / Principles of Integrl Evlution C FOCUS ON CONCEPTS 57. Sketch the region whose re is + nd use our sketch to show tht + = d 58. () Give resonle informl rgument, sed on res, tht eplins wh the integrls diverge. Show tht sin cos nd diverges. cos 59. In electromgnetic theor, the mgnetic potentil t point on the is of circulr coil is given u = πni r k (r + ) 3/ where N,I,r,k, nd re constnts. Find u. 6. The verge speed, v, of the molecules of n idel gs is given v = 4 ( ) M 3/ v 3 e Mv /(RT ) dv π RT nd the root-men-squre speed, v rms, vrms = 4 ( ) M 3/ v 4 e Mv /(RT ) dv π RT where v is the moleculr speed, T is the gs temperture, M is the moleculr weight of the gs, nd R is the gs constnt. () Use CAS to show tht 3 e =, > 4 nd use this result to show tht v = 8RT /(πm). Use CAS to show tht 4 e = 3 π 8, > 5 nd use this result to show tht v rms = 3RT /M. 6. In Eercise 5 of Section 6.6, we determined the work required to lift 6 l stellite to n oritl position tht is mi ove the Erth s surfce. The ides discussed in tht eercise will e needed here. () Find definite integrl tht represents the work required to lift 6 l stellite to position miles ove the Erth s surfce. Find definite integrl tht represents the work required to lift 6 l stellite n infinite distnce ove the Erth s surfce. Evlute the integrl. Note: The result otined here is sometimes clled the work required to escpe the Erth s grvit.] C 6 63 A trnsform is formul tht converts or trnsforms one function into nother. Trnsforms re used in pplictions to convert difficult prolem into n esier prolem whose solution cn then e used to solve the originl difficult prolem. The Lplce trnsform of function f(t),which pls n importnt role in the stud of differentil equtions, is denoted {f(t)} nd is defined {f(t)}= e st f(t)dt In this formul s is treted s constnt in the integrtion process; thus, the Lplce trnsform hs the effect of trnsforming f(t)into function of s. Use this formul in these eercises. 6. Show tht () {} = s, s> {et }= s, s> (c) {sin t} = s +, s> (d) {cos t} = s s +, s>. 63. In ech prt, find the Lplce trnsform. () f(t) = { t, s> f(t) = t, s>, t < 3 (c) f(t) =, t 3, s > 64. Lter in the tet, we will show tht e = π Confirm tht this is resonle using CAS or clcultor with numericl integrtion cpilit. 65. Use the result in Eercise 64 to show tht π () e =, > + e /σ =, σ>. πσ A convergent improper integrl over n infinite intervl cn e pproimted first replcing the infinite limit(s) of integrtion finite limit(s), then using numericl integrtion technique, such s Simpson s rule, to pproimte the integrl with finite limit(s). This technique is illustrted in these eercises. 66. Suppose tht the integrl in Eercise 64 is pproimted first writing it s e = K e + K e then dropping the second term, nd then ppling Simpson s rule to the integrl K e The resulting pproimtion hs two sources of error: the error from Simpson s rule nd the error E = K e (cont.)

11 tht results from discrding the second term. We cll E the trunction error. () Approimte the integrl in Eercise 64 ppling Simpson s rule with n = sudivisions to the integrl 3 e Round our nswer to four deciml plces nd compre it to π rounded to four deciml plces. Use the result tht ou otined in Eercise 5 nd the fct tht e 3 e for 3 to show tht the trunction error for the pproimtion in prt () stisfies <E< () It cn e shown tht 6 + = π 3 Approimte this integrl ppling Simpson s rule with n = sudivisions to the integrl Round our nswer to three deciml plces nd compre it to π/3 rounded to three deciml plces. Use the result tht ou otined in Eercise 5 nd the fct tht /( 6 + ) </ 6 for 4 to show tht the trunction error for the pproimtion in prt () stisfies <E< For wht vlues of p does e p converge? 69. Show tht p. / p converges if p< nd diverges if C 7. Chpter 7 Review Eercises 557 It is sometimes possile to convert n improper integrl into proper integrl hving the sme vlue mking n pproprite sustitution. Evlute the following integrl mking the indicted sustitution, nd investigte wht hppens if ou evlute the integrl directl using CAS. + ; u = 7 7 Trnsform the given improper integrl into proper integrl mking the stted u-sustitution; then pproimte the proper integrl Simpson s rule with n = sudivisions. Round our nswer to three deciml plces. cos 7. ; u = 7. sin ; u = 73. Writing Wht is improper out n integrl over n infinite intervl? Eplin wh Definition 5.5. for f() fils for f(). Discuss strteg for ssigning vlue to f(). 74. Writing Wht is improper out definite integrl over n intervl on which the integrnd hs n infinite discontinuit? Eplin wh Definition 5.5. for f() fils if the grph of f hs verticl smptote t =. Discuss strteg for ssigning vlue to f() in this circumstnce. QUICK CHECK ANSWERS 7.8. () proper improper, since cot hs n infinite discontinuit t = π (c) improper, since there is n infinite intervl of integrtion (d) improper, since there is n infinite intervl of integrtion nd the integrnd hs n infinite discontinuit t =. lim cot (c) lim (d) π π/4 + + lim + lim p ; p> 4. () diverges (c) diverges (d) 3 CHAPTER 7 REVIEW EXERCISES 6 Evlute the given integrl with the id of n pproprite u-sustitution sec π cos 3. sin 4. ln 9 5. tn ( ) sec ( ) () Evlute the integrl three ws: using the sustitution u =, using the sustitution u =, nd completing the squre. Show tht the nswers in prt () re equivlent.

12 558 Chpter 7 / Principles of Integrl Evlution 3 8. Evlute the integrl + () using integrtion prts using the sustitution u = +. 9 Use integrtion prts to evlute the integrl. 9. e. sin. ln( + 3). / tn () 3. Evlute 8 4 cos using tulr integrtion prts. 4. A prticle moving long the -is hs velocit function v(t) = t e t. How fr does the prticle trvel from time t = tot = 5? 5 Evlute the integrl. 5. sin 5θdθ 6. sin 3 cos sin cos 8. sin 4. π/6 sin cos 4 cos 5 ( ) 6 Evlute the integrl mking n pproprite trigonometric sustitution Evlute the integrl using the method of prtil frctions ( )( 3) ( + ) Consider the integrl 3. () Evlute the integrl using the sustitution = sec θ. For wht vlues of is our result vlid? Evlute the integrl using the sustitution = sin θ. For wht vlues of is our result vlid? (c) Evlute the integrl using the method of prtil frctions. For wht vlues of is our result vlid? 34. Find the re of the region tht is enclosed the curves = ( 3)/( 3 + ), =, =, nd = Use the Endpper Integrl Tle to evlute the integrl. 35. sin 7 cos ( 3 )e tn Approimte the integrl using () the midpoint pproimtion M, the trpezoidl pproimtion T, nd (c) Simpson s rule pproimtion S. In ech cse, find the ect vlue of the integrl nd pproimte the solute error. Epress our nswers to t lest four deciml plces Use inequlities (), (3), nd (4) of Section 7.7 to find upper ounds on the errors in prts (),, or (c) of the indicted eercise. 43. Eercise Eercise Use inequlities (), (3), nd (4) of Section 7.7 to find numer n of suintervls for () the midpoint pproimtion M n, the trpezoidl pproimtion T n, nd (c) Simpson s rule pproimtion S n to ensure the solute error will e less thn Eercise Eercise Evlute the integrl if it converges e Find the re tht is enclosed etween the -is nd the curve = (ln )/ for e. 5. Find the volume of the solid tht is generted when the region etween the -is nd the curve = e for is revolved out the -is. 53. Find positive vlue of tht stisfies the eqution + = 54. Consider the following methods for evluting integrls: u-sustitution, integrtion prts, prtil frctions, reduction formuls, nd trigonometric sustitutions. In ech prt, stte the pproch tht ou would tr first to evlute the integrl. If none of them seems pproprite, then s so. Youneed not evlute the integrl. () sin cos sin (cont.)

13 (c) (e) (g) (i) tn 7 (d) (f ) tn (h) 4 tn 7 sec 3 ( + ) Evlute the integrl cos 3 (3 + ) 3/ π/4 57. tn 7 cos θ θdθ 58. sin θ 6 sin θ + dθ sin cos 3 6. ( 3) / 6. e cos 3 6. ( ) 3/ / Chpter 7 Mking Connections 559 /3 64. ( )( + )( 3) e / sin ( + ) +,, > ln 7. (4 9 ) e ( + + ) tn 5 4 sec 4 4 sec θ tn 3 θ tn θ dθ CHAPTER 7 MAKING CONNECTIONS C CAS. Recll from Theorem 3.3. nd the discussion preceding it tht if f () >, then the function f is incresing nd hs n inverse function. Prts (),, nd (c) of this prolem show tht if this condition is stisfied nd if f is continuous, then definite integrl of f cn e epressed in terms of definite integrl of f. () Use integrtion prts to show tht f() = f f() f () Use the result in prt () to show tht if = f(), then f() = f f() f f() f () d (c) Show tht if we let α = f() nd β = f, then the result in prt cn e written s β α f () = βf (β) αf (α) f (β) f (α) f(). In ech prt, use the result in Eercise to otin the eqution, nd then confirm tht the eqution is correct performing the integrtions. / () sin = ( ) π/6 sin sin e e ln = (e e) e C 3. The Gmm function, Ɣ(), is defined s Ɣ() = t e t dt It cn e shown tht this improper integrl converges if nd onl if >. () Find Ɣ(). Prove: Ɣ( + ) = Ɣ() for ll >. Hint: Use integrtion prts.] (c) Use the results in prts () nd to find Ɣ(), Ɣ(3), nd Ɣ(4); nd then mke conjecture out Ɣ(n) for positive integer vlues of n. (d) Show tht Ɣ ( ) = π.hint: See Eercise 64 of Section 7.8.] (e) Use the results otined in prts nd (d) to show tht Ɣ ( ) 3 = ( π nd Ɣ 5 ) = 3 4 π. 4. Refer to the Gmm function defined in Eercise 3 to show tht () (ln ) n = ( ) n Ɣ(n + ), n> Hint: Let t = ln.] ( ) n + e n = Ɣ, n>. n Hint: Let t = n. Use the result in Eercise 3.] 5. A simple pendulum consists of mss tht swings in verticl plne t the end of mssless rod of length L, s shown in the ccompning figure. Suppose tht simple pendulum is displced through n ngle θ nd relesed from rest. It cn e

14 56 Chpter 7 / Principles of Integrl Evlution shown tht in the sence of friction, the time T required for the pendulum to mke one complete ck-nd-forth swing, clled the period, is given T = 8L g θ cos θ cos θ dθ () where θ = θ(t) is the ngle the pendulum mkes with the verticl t time t. The improper integrl in () is difficult to evlute numericll. B sustitution outlined elow it cn e shown tht the period cn e epressed s T = 4 L g π/ dφ () k sin φ where k = sin(θ /). The integrl in () is clled complete elliptic integrl of the first kind nd is more esil evluted numericl methods. () Otin () from () sustituting cos θ = sin (θ/) cos θ = sin (θ /) k = sin(θ /) nd then mking the chnge of vrile sin φ = sin(θ /) sin(θ /) = sin(θ /) k Use () nd the numericl integrtion cpilit of our CAS to estimte the period of simple pendulum for which L =.5 ft, θ =, nd g = 3 ft/s. u L Figure E-5 E XPANDING THE C ALCULUS H ORIZON To lern how numericl integrtion cn e pplied to the cost nlsis of n engineering project, see the module entitled Rilrod Design t:

The Trapezoidal Rule

The Trapezoidal Rule _.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

More information

Improper Integrals with Infinite Limits of Integration

Improper Integrals with Infinite Limits of Integration 6_88.qd // : PM Pge 578 578 CHAPTER 8 Integrtion Techniques, L Hôpitl s Rule, nd Improper Integrls Section 8.8 f() = d The unounded region hs n re of. Figure 8.7 Improper Integrls Evlute n improper integrl

More information

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ). AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

More information

Review Exercises for Chapter 4

Review Exercises for Chapter 4 _R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises

More information

Chapter 6 Techniques of Integration

Chapter 6 Techniques of Integration MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln

More information

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral. Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

More information

4.6 Numerical Integration

4.6 Numerical Integration .6 Numericl Integrtion 5.6 Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll

More information

The Trapezoidal Rule

The Trapezoidal Rule SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion Approimte

More information

LINEAR ALGEBRA APPLIED

LINEAR ALGEBRA APPLIED 5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order

More information

Section 6.1 INTRO to LAPLACE TRANSFORMS

Section 6.1 INTRO to LAPLACE TRANSFORMS Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

More information

5.7 Improper Integrals

5.7 Improper Integrals 458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the

More information

Chapter 7: Applications of Integrals

Chapter 7: Applications of Integrals Chpter 7: Applictions of Integrls 78 Chpter 7 Overview: Applictions of Integrls Clculus, like most mthemticl fields, egn with tring to solve everd prolems. The theor nd opertions were formlized lter. As

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

What Is Calculus? 42 CHAPTER 1 Limits and Their Properties

What Is Calculus? 42 CHAPTER 1 Limits and Their Properties 60_00.qd //0 : PM Pge CHAPTER Limits nd Their Properties The Mistress Fellows, Girton College, Cmridge Section. STUDY TIP As ou progress through this course, rememer tht lerning clculus is just one of

More information

Section 4: Integration ECO4112F 2011

Section 4: Integration ECO4112F 2011 Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

M344 - ADVANCED ENGINEERING MATHEMATICS

M344 - ADVANCED ENGINEERING MATHEMATICS M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If

More information

Fundamental Theorem of Calculus

Fundamental Theorem of Calculus Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under

More information

5.4. The Fundamental Theorem of Calculus. 356 Chapter 5: Integration. Mean Value Theorem for Definite Integrals

5.4. The Fundamental Theorem of Calculus. 356 Chapter 5: Integration. Mean Value Theorem for Definite Integrals 56 Chter 5: Integrtion 5.4 The Fundmentl Theorem of Clculus HISTORICA BIOGRAPHY Sir Isc Newton (64 77) In this section we resent the Fundmentl Theorem of Clculus, which is the centrl theorem of integrl

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

APPLICATIONS OF DEFINITE INTEGRALS

APPLICATIONS OF DEFINITE INTEGRALS Chpter 6 APPICATIONS OF DEFINITE INTEGRAS OVERVIEW In Chpter 5 we discovered the connection etween Riemnn sums ssocited with prtition P of the finite closed intervl [, ] nd the process of integrtion. We

More information

Math& 152 Section Integration by Parts

Math& 152 Section Integration by Parts Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible

More information

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

More information

Time in Seconds Speed in ft/sec (a) Sketch a possible graph for this function.

Time in Seconds Speed in ft/sec (a) Sketch a possible graph for this function. 4. Are under Curve A cr is trveling so tht its speed is never decresing during 1-second intervl. The speed t vrious moments in time is listed in the tle elow. Time in Seconds 3 6 9 1 Speed in t/sec 3 37

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

The practical version

The practical version Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht

More information

5.1 Estimating with Finite Sums Calculus

5.1 Estimating with Finite Sums Calculus 5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during

More information

10.2 The Ellipse and the Hyperbola

10.2 The Ellipse and the Hyperbola CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point

More information

1 Part II: Numerical Integration

1 Part II: Numerical Integration Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble

More information

Total Score Maximum

Total Score Maximum Lst Nme: Mth 8: Honours Clculus II Dr. J. Bowmn 9: : April 5, 7 Finl Em First Nme: Student ID: Question 4 5 6 7 Totl Score Mimum 6 4 8 9 4 No clcultors or formul sheets. Check tht you hve 6 pges.. Find

More information

( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.

( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists. AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find

More information

Calculus AB. For a function f(x), the derivative would be f '(

Calculus AB. For a function f(x), the derivative would be f '( lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:

More information

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx... Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

More information

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

Section - 2 MORE PROPERTIES

Section - 2 MORE PROPERTIES LOCUS Section - MORE PROPERTES n section -, we delt with some sic properties tht definite integrls stisf. This section continues with the development of some more properties tht re not so trivil, nd, when

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

More information

and that at t = 0 the object is at position 5. Find the position of the object at t = 2.

and that at t = 0 the object is at position 5. Find the position of the object at t = 2. 7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we

More information

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

More information

Introduction to Algebra - Part 2

Introduction to Algebra - Part 2 Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round

More information

Mathematics Number: Logarithms

Mathematics Number: Logarithms plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement

More information

Chapter 8.2: The Integral

Chapter 8.2: The Integral Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in

More information

Topics Covered AP Calculus AB

Topics Covered AP Calculus AB Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

More information

Interpreting Integrals and the Fundamental Theorem

Interpreting Integrals and the Fundamental Theorem Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

More information

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic

More information

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into

More information

Section 6.1 Definite Integral

Section 6.1 Definite Integral Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

More information

x ) dx dx x sec x over the interval (, ).

x ) dx dx x sec x over the interval (, ). Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor

More information

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2. Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot

More information

10 Vector Integral Calculus

10 Vector Integral Calculus Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve

More information

Arc Length and Surfaces of Revolution. Find the arc length of a smooth curve. Find the area of a surface of revolution. <...

Arc Length and Surfaces of Revolution. Find the arc length of a smooth curve. Find the area of a surface of revolution. <... 76 CHAPTER 7 Applictions of Integrtion The Dutch mthemticin Christin Hugens, who invented the pendulum clock, nd Jmes Gregor (6 675), Scottish mthemticin, oth mde erl contriutions to the prolem of finding

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

2.4 Linear Inequalities and Interval Notation

2.4 Linear Inequalities and Interval Notation .4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

More information

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a). The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

More information

7. Indefinite Integrals

7. Indefinite Integrals 7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find

More information

Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION

Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..

More information

sec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5

sec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5 Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

More information

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

More information

Chapter 1: Logarithmic functions and indices

Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

More information

Chapter 3 Exponential and Logarithmic Functions Section 3.1

Chapter 3 Exponential and Logarithmic Functions Section 3.1 Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re non-lgebric functions. The re clled trnscendentl functions. The eponentil

More information

Section 6.1 INTRO to LAPLACE TRANSFORMS

Section 6.1 INTRO to LAPLACE TRANSFORMS Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

More information

Unit 5. Integration techniques

Unit 5. Integration techniques 18.01 EXERCISES Unit 5. Integrtion techniques 5A. Inverse trigonometric functions; Hyperbolic functions 5A-1 Evlute ) tn 1 3 b) sin 1 ( 3/) c) If θ = tn 1 5, then evlute sin θ, cos θ, cot θ, csc θ, nd

More information

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 + Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

More information

7.2 The Definite Integral

7.2 The Definite Integral 7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

More information

SECTION 9-4 Translation of Axes

SECTION 9-4 Translation of Axes 9-4 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.

More information

5.1 How do we Measure Distance Traveled given Velocity? Student Notes

5.1 How do we Measure Distance Traveled given Velocity? Student Notes . How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the x-xis & y-xis

More information

Topic 1 Notes Jeremy Orloff

Topic 1 Notes Jeremy Orloff Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

More information

Chapter 3 Single Random Variables and Probability Distributions (Part 2)

Chapter 3 Single Random Variables and Probability Distributions (Part 2) Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their

More information

An Overview of Integration

An Overview of Integration An Overview of Integrtion S. F. Ellermeyer July 26, 2 The Definite Integrl of Function f Over n Intervl, Suppose tht f is continuous function defined on n intervl,. The definite integrl of f from to is

More information

ES.182A Topic 32 Notes Jeremy Orloff

ES.182A Topic 32 Notes Jeremy Orloff ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In

More information

Math 1431 Section 6.1. f x dx, find f. Question 22: If. a. 5 b. π c. π-5 d. 0 e. -5. Question 33: Choose the correct statement given that

Math 1431 Section 6.1. f x dx, find f. Question 22: If. a. 5 b. π c. π-5 d. 0 e. -5. Question 33: Choose the correct statement given that Mth 43 Section 6 Question : If f d nd f d, find f 4 d π c π- d e - Question 33: Choose the correct sttement given tht 7 f d 8 nd 7 f d3 7 c d f d3 f d f d f d e None of these Mth 43 Section 6 Are Under

More information

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

More information

0.1 THE REAL NUMBER LINE AND ORDER

0.1 THE REAL NUMBER LINE AND ORDER 6000_000.qd //0 :6 AM Pge 0-0- CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.

More information

x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b

x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick

More information

The Evaluation Theorem

The Evaluation Theorem These notes closely follow the presenttion of the mteril given in Jmes Stewrt s textook Clculus, Concepts nd Contexts (2nd edition) These notes re intended primrily for in-clss presenttion nd should not

More information

C Precalculus Review. C.1 Real Numbers and the Real Number Line. Real Numbers and the Real Number Line

C Precalculus Review. C.1 Real Numbers and the Real Number Line. Real Numbers and the Real Number Line C. Rel Numers nd the Rel Numer Line C C Preclculus Review C. Rel Numers nd the Rel Numer Line Represent nd clssif rel numers. Order rel numers nd use inequlities. Find the solute vlues of rel numers nd

More information

Chapter 9. Arc Length and Surface Area

Chapter 9. Arc Length and Surface Area Chpter 9. Arc Length nd Surfce Are In which We ppl integrtion to stud the lengths of curves nd the re of surfces. 9. Arc Length (Tet 547 553) P n P 2 P P 2 n b P i ( i, f( i )) P i ( i, f( i )) distnce

More information

y = f(x) This means that there must be a point, c, where the Figure 1

y = f(x) This means that there must be a point, c, where the Figure 1 Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile

More information

AP Calculus AB Unit 5 (Ch. 6): The Definite Integral: Day 12 Chapter 6 Review

AP Calculus AB Unit 5 (Ch. 6): The Definite Integral: Day 12 Chapter 6 Review AP Clculus AB Unit 5 (Ch. 6): The Definite Integrl: Dy Nme o Are Approximtions Riemnn Sums: LRAM, MRAM, RRAM Chpter 6 Review Trpezoidl Rule: T = h ( y + y + y +!+ y + y 0 n n) **Know how to find rectngle

More information

AB Calculus Review Sheet

AB Calculus Review Sheet AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

More information

P 1 (x 1, y 1 ) is given by,.

P 1 (x 1, y 1 ) is given by,. MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce

More information

Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED

Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type

More information

Linear Inequalities. Work Sheet 1

Linear Inequalities. Work Sheet 1 Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

More information

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests. ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

More information

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one

More information

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

A. Limits - L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. -1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1.

A. Limits - L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. -1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1. A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( ) lim where lim f or lim f limg. c g = c limg( ) = c = c = c How to find it: Try nd find limits by

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230 Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

More information

Math 190 Chapter 5 Lecture Notes. Professor Miguel Ornelas

Math 190 Chapter 5 Lecture Notes. Professor Miguel Ornelas Mth 19 Chpter 5 Lecture Notes Professor Miguel Ornels 1 M. Ornels Mth 19 Lecture Notes Section 5.1 Section 5.1 Ares nd Distnce Definition The re A of the region S tht lies under the grph of the continuous

More information

13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes

13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce

More information

( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht

More information

Logarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.

Logarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100. Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the

More information