5: The Definite Integral


 Sheena Grant
 1 years ago
 Views:
Transcription
1 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce tht the oject hs ( ) trveled during the first four seconds? The old d = r t only wors for constnt speeds Divide nd Conquer Let's te cue from the movies! As you now, movie is just series of still imges, tht re run relly fst. Ech imge is still (no movement), ut series of them give the illusion of movement. Perhps we could te our velocity eqution, chop it up into series of still imges, nd get the motion from tht Let's sy it this wy: for ny short intervl of time, the chnge in velocity is not very lrge thus, for short intervl of time, we could consider the oject to e moving with constnt velocity, nd use our old fmilir distnce eqution. There is grphic interprettion of this ide creting rectngles round the grph of ( ) We'll strt y chopping the time intervl into four pieces: v t. Notice tht the oject is t rest round time.5 seconds thus, it turns round nd heds c in the other direction. Thus, the totl distnce trveled will e different from the net distnce trveled (displcement). Keep this in mind for lter! The totl re of these rectngles is 5 the totl distnce trveled is 5 meters. This is only n pproximtion, though we're supposed to e chopping the time intervl into sections where the velocity is lmost constnt! Let's do etter: try ten intervls. Tht gives totl re of This should e little etter however, there will still e some error! Let's try once gin, with intervls: HOLLOMAN S AP CALCULUS AB ABC NOTES 5, PAGE OF
2 This gives n re of 4.8 this is etter, ut still not the nswer. Reductio d infinitum The nswer requires tht we consider n infinite numer of intervls! However, we'll leve tht for it, since tht requires limits. Let's te one step t time. Three Pths to the Summit The pictures tht I drew ove ll hve rectngles tht hve the left side on the function we cll these lefthnd rectngles. You could lso use midpoints (the height of the rectngle is t the midpoint of the top edge) or right sides (te wild guess!). You'll get slightly different nswers ut tht's oy; once we wor towrds the nswer, they'll ll come out the sme. Exmples [.] Estimte the re under the curve rectngles. y= e x in the intervl [, ] with four nd ten First, with four rectngles: The estimted re is.9. Now with ten rectngles: HOLLOMAN S AP CALCULUS AB ABC NOTES 5, PAGE OF
3 The estimted re is.66. [.] The following dt were collected from moving vehicle: time (sec) velocity (ft/sec) Estimte the distnce trveled y the vehicle during the thirty second intervl. Note tht the dt hve lredy divided the intervl into prtition! The width of ech suintervl is 5 (seconds), nd the height of the rectngle is the given velocity (feet per second). width height re This gives totl re of 4 which mens totl distnce of 4 feet. 5.: Definite Integrls The Generic Pth: Riemnn Sums Let's generlize the ides we explored in the previous section A prtition of the intervl [, ] is set of vlues ( x i ) so tht < x < x <... < xn <. If we =, then P= { x, x, x... x n } is prtition of the intervl [, ]. cll x = nd xn The th suintervl of P is [ x x ], e the width of the th suintervl of P. x. Let c e ny vlue in the th suintervl of P. Also, let If f is function which is continuous on the intervl [, ], then the Riemnn sum of f (on tht intervl) is f ( c ) n = x. Note tht this is the sum of unch of rectngle res The Definite Integrl Bc when we were developing the ide of derivtives, we creted generic point, nd then moved the point ritrrily close to the trget point we llowed the distnce etween the points to pproch zero. This limiting process gve us the slope of the curve. Now consider the rectngles res from the Riemnn sum. One might thin tht dding more points to the prtition might me the rectngle res get relly smll, nd dd up to the re HOLLOMAN S AP CALCULUS AB ABC NOTES 5, PAGE OF
4 under the curve however, just dding more points doesn't me ll of the rectngles smll. To do tht, you hve to dd points so tht the lrgest x pproches zero (the lrgest x is clled the norm of P, or P ). n If lim f ( c) x exists, then we sy tht f is integrle over the intervl [, ] P = vlue of the limit is clled the definite integrl of f over [, ] Nottion., nd the Tht limit nottion is terrile difficult to write, so we crete new nottion for the definite integrl of f over [, ] : ( ) f x dx. This is red s "the integrl of f from to." Evluting Definite Integrls Definite integrls re defined in such wy tht they pproximte the re etween curve nd the xxis (if it lies ove the xxis). However, since this is mth (nd not relity), the re etween the curve nd the xxis for functions which lie elow the xxis is negtive. Thus, the, is the re ove the xis plus the re elow the xis. definite integrl of f over [ ] If you relly wnt re (rel, positive re), then it's re ove the xis minus re elow the xis. Returning to our velocity exmple from while c: the totl distnce trveled will e the re etween the line v( t) = 9.8t+ 5 nd the xxis. Fortuntely, this is just pir of tringles! First, note tht v (.55) =. The re of the left/upper tringle is ( 5 )(.55 ) =.888, nd the re of the right/lower tringle is ( 4.)( 4.55) =.88, for totl re of 4.76 ( totl distnce of 4.76 meters). The displcement is =.6 meters. Of course, rther thn doing ll of this y hnd, you could use your clcultor note tht the clcultor will do the definite integrl; not the re under the curve. My clcultor gives the vlue of the definite integrl t.6. Exmples 7 [.] Find dx. This is rectngle with se length of 8 nd height of ; the re (nd the vlue of the definite integrl) is 6. 5 [4.] Find ( ) 5 x dx. 5 =. This is tringle with se 5 nd height 5; the re is ( 5)( 5) HOLLOMAN S AP CALCULUS AB ABC NOTES 5, PAGE 4 OF
5 [5.] Find 4 x dx. This is semicircle of rdius. The re is 4π. 5.: Definite Integrls nd Antiderivtives Properties of the Definite Integrl Reversing the limits negtes the integrl: ( ) = ( ) f x dx f x dx The integrl of point is zero: f ( x ) dx= Constnts cn e fctored out: ( ) = ( ) f x dx f x dx Integrls cn e "distriuted" cross sums nd differences: ( ) ± ( ) = ( ) ± ( ) f x g x dx f x dx g x dx c c Integrls over djcent intervls cn e comined: ( ) + ( ) = ( ) f x dx f x dx f x dx The Averge Vlue of Function We ll now how to find n verge dd up ll the vlues, nd divide y the numer of vlues. It turns out tht you cn do tht with n infinite numer of things, lso! Adding n infinite numer of things cretes n integrl, though. The verge vlue of f ( x ) over the intervl [, ] is ( ) f x dx. The Men Vlue Theorem (for Definite Integrls) There is some point ( x= c ) in the intervl [, ] so tht f ( c) = f ( x) dx. The Men Vlue Theorem for Derivtives sys tht for ny intervl, there is some point where the slope of the tngent equls the slope of the secnt. The Men Vlue Theorem for Integrls sys tht for ny intervl, there is some point where the re of rectngle equls the re under the curve (( ) ( ) ( ) f c = f x dx ). The Connection: A First Glimpse The derivtive is found with quotient of infinitely smll differences. The definite integrl is found through sum of infinitely smll products (res). Sums nd differences quotients nd products surely there is some connection! nd there is. Here's loo hed which you'll need for few of the prolems in this section. ( ) = ( ) ( ), where F( x ) is ny ntiderivtive of ( ) f x dx F F f x. HOLLOMAN S AP CALCULUS AB ABC NOTES 5, PAGE 5 OF
6 Exmples [6.] If f ( x) dx= 7 nd g( x) dx=6, then find ( ) + ( ) Using the properties of integrls, ( ) ( ) ( ) ( ) ( ) ( ) f x g x dx. f x + g x dx= f x dx+ g x dx= = [7.] Find ( ) x x dx. Since I don't now hndy formul for the re under prol, I'll use ntiderivtives. One f x = x x is F( x) = x x x. Thus, ( ) ( ) ( ) f x dx= F F F = + =, so ( ) 6 6 x x dx= = = ntiderivtive of ( ) F ( ) = = nd ( ) [8.] Find the verge vlue of on the intervl [, 4 ]. x 4 The formul gives me dx 4. I'll hve to use n ntiderivtive gin how out x x? 4 4 dx= x = = = x : The Fundmentl Theorem of Clculus FTC, Prt x d If g( x) = f ( t) dt, then g ( x ) f ( x ) dx =. In other words, the derivtive of n integrl is just the plin function. It's s if the derivtive nd the integrl re inverse opertions The hirs on the c of your nec should e stnding on end The Integrl s Function x The ide of g( x) ( ) = f t dt needs some dditionl investigtion there re lots of potentil questions tht cn e sed out functions tht re defined s integrls of other functions! Here re just few exmples from single scenrio The grph of f ( t ) is shown elow. It consists of semicircle nd line segment. HOLLOMAN S AP CALCULUS AB ABC NOTES 5, PAGE 6 OF
7 = f t dt. x Define A( x) ( ) A ( 6) is the re under ( ) circle. ( ) ( ) π A = π = = 9π. 4 4 A ( 8) is the re under ( ) tringle! ( ) ( ) ( )( ) f t etween x= nd 6 f t etween x= nd 8 A 8 = π = 8π x= ( ) f t dt which is qurter HOLLOMAN S AP CALCULUS AB ABC NOTES 5, PAGE 7 OF 8 x= : ( ) f t dt. This is semicircle nd Note tht A( x ) is n incresing function s x gets lrger, the mount of re under f ( t ) eeps getting lrger. = 8 =π+. 8 The verge vlue of f ( t ) on the intervl [,8 ] is f ( t) dt A( ) FTC, Prt 8 8 If f ( x ) is ny ntiderivtive of g( x ), then g ( x ) dx= f ( ) f ( ). Note tht f ( x ) eing n ntiderivtive of g( x ) mens tht f ( x) g( x) =. Thus, this theorem is ting the integrl of n ntiderivtive. Also note tht the integrl of this derivtive resulted in the originl function (sort of). Finding Are If you wnt to use definite integrls to find the re etween curve nd the xxis, rememer tht res elow the xis will e negtive! One wy to find the totl re is to first determine where the function is equl to zero this will divide the res into ove/elow the xxis. Add the integrls from the sections ove the xis, nd sutrct the integrls from the sections elow the xis. Another wy to do this is to just find the re under the solute vlue of the function. The solute vlue will simply force ll of the res to e positive. This mes things relly esy especilly if you re using your clcultor Exmples π cos x dx. [9.] Find ( ) Well n ntiderivtive of cos( x ) is ( ) π ( x ) dx ( x π ) ( ) ( ) etween sin( x ) nd the xxis is zero! sin x so cos = sin ] = sin π sin = =. Note tht this does not men tht the re [.] Find the totl re etween f ( x) = x x nd the xxis on the intervl [,] Let's e creful nd loo t the grph..
8 We'll need to split this up! The totl re will e ( ) ( ) ( ) ( ) x x dx+ x x dx x x dx+ x x dx 4 An ntiderivtive of f ( x) = x x is F( x) x x =. Tht gives us 4 F ( x ) F ( x ) F ( x ) F ( x ) F( ) F( ) F( ) F( ) F( ) F( ) F( ) F( ) F( ) F( ) F( ) F( ) F( ) F( ) F( ) F( ) ( ) ( ) + ( ) ( ) + ( ) F F F F F Let's strt plugging in! F( ) = 6 4= 4 = 4 F( ) = = 4 4 F = ( ) F ( ) = = 4 4 F ( ) = 6 4= 4 = = = So finlly, we get ( ) d t dt dx. π tn x. x [.] Find tn( ) Esy! The derivtive is ( ) 5.5: The Trpezoidl Rule Another Pth Now tht we hve firm connection etween integrls nd re, let's pproch the re ide gin. Before, we used rectngles to pproximte the re under the curve simply ecuse HOLLOMAN S AP CALCULUS AB ABC NOTES 5, PAGE 8 OF
9 rectngles re very simple geometric figures. Als, they don't relly do such good jo of pproximting re under the curve until you use lot of very tiny rectngles. There is nother shpe which does etter jo the trpezoid! Divide the intervl in question s you would efore, ut now connect the function vlues to otin trpezoids. For our liner exmple (wy c t the eginning of this chpter), we get n exct nswer with s few s two trpezoids! Let's te the function f ( x) = 4 ( x ) on the intervl [ ] exmple. Here is the pproximte re with four rectngles:, 4 with four regions s n (tht's n re of 9.848) Here is the pproximte re with four trpezoids: (tht's n re of 8.788) The ctul re is 9. You cn see tht the trpezoids do much etter jo of the pproximtion! Simpson's Rule/Approximtion You cn do even etter with your pproximtion if, insted of trpezoids, you use rectngles cpped with little prolic rches. It turns out tht there is (reltively simple) formul for this nd (firly simple) formul tht generlizes the entire procedure! It's clled Simpson's Rule: To pproximte f ( x ) dx, divide the intervl [, ] into n even numer of suintervls y o+ y + y + y + + y y y n + n + n. n Hppily, this topic is not prt of the AP Exm! (me n even) nd then clculte ( ) HOLLOMAN S AP CALCULUS AB ABC NOTES 5, PAGE 9 OF
10 Exmples [.] Use four trpezoids to estimte the vlue of Let's loo t the picture: x dx. The res of the trpezoids re + = = = + = = = + = = = + 8 = = 8 8 This gives totl re of = 6 = 4.5. (the ctul re is 4) HOLLOMAN S AP CALCULUS AB ABC NOTES 5, PAGE OF
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 information5.1 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 xxis & yxis
More informationDefinite 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 lefthnd
More informationSuppose 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 informationSuppose 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 informationInterpreting 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 informationSection 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 informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationThe area under the graph of f and above the xaxis 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 xxis etween nd is denoted y f(x) dx nd clled the
More informationMA123, 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. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More information7.1 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 information2 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 informationProperties 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 informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More information7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?
7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge
More informationSection 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 informationGoals: 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 informationTopics 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 informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More information( ) 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( ) 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 informationChapter 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 informationTime 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 1second 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 informationAP 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 informationBob Brown Math 251 Calculus 1 Chapter 5, Section 4 Completed 1 CCBC Dundalk
Bo Brown Mth Clculus Chpter, Section Completed CCBC Dundlk The Fundmentl Theorem of Clculus Informlly, the Fundmentl Theorem of Clculus (FTC) sttes tht differentition nd definite integrtion re inverse
More informationBob Brown Math 251 Calculus 1 Chapter 5, Section 4 1 CCBC Dundalk
Bo Brown Mth Clculus Chpter, Section CCBC Dundlk The Fundmentl Theorem of Clculus Informlly, the Fundmentl Theorem of Clculus (FTC) sttes tht differentition nd definite integrtion re inverse opertions
More informationUnit Six AP Calculus Unit 6 Review Definite Integrals. Name Period Date NONCALCULATOR SECTION
Unit Six AP Clculus Unit 6 Review Definite Integrls Nme Period Dte NONCALCULATOR SECTION Voculry: Directions Define ech word nd give n exmple. 1. Definite Integrl. Men Vlue Theorem (for definite integrls)
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationf(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as f(x) dx = lim f(x i ) x; i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationTime 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 1second 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 informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationAB 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 information2.4 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 informationSection Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled?
Section 5.  Ares nd Distnces Exmple : Suppose cr trvels t constnt 5 miles per hour for 2 hours. Wht is the totl distnce trveled? Exmple 2: Suppose cr trvels 75 miles per hour for the first hour, 7 miles
More informationBig idea in Calculus: approximation
Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More information5.1 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 informationF is n ntiderivtive èor èindeæniteè integrlè off if F 0 èxè =fèxè. Nottion: F èxè = ; it mens F 0 èxè=fèxè ëthe integrl of f of x dee x" Bsic list: xn
Mth 70 Topics for third exm Chpter 3: Applictions of Derivtives x7: Liner pproximtion nd diæerentils Ide: The tngent line to grph of function mkes good pproximtion to the function, ner the point of tngency.
More informationCh AP Problems
Ch. 7.7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationChapter 8.2: The Integral
Chpter 8.: The Integrl You cn think of Clculus s doulewide 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 informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationAn 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 informationThe Fundamental Theorem of Calculus, Particle Motion, and Average Value
The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt
More informationMath 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 informationThe 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 informationThe 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 informationThe Fundamental Theorem of Calculus
The Fundmentl Theorem of Clculus MATH 151 Clculus for Mngement J. Robert Buchnn Deprtment of Mthemtics Fll 2018 Objectives Define nd evlute definite integrls using the concept of re. Evlute definite integrls
More informationReview 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 informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationIntegrals  Motivation
Integrls  Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is nonliner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but
More informationPolynomials 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 informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationImproper 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 informationFundamental 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 information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationCalculus 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 informationMath 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n
More information( ) Same as above but m = f x = f x  symmetric to yaxis. 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 informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose ntiderivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
More information1 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 informationThe 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 information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Tody we provide the connection
More informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
More informationMath 116 Calculus II
Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationLab 11 Approximate Integration
Nme Student ID # Instructor L Period Dte Due L 11 Approximte Integrtion Ojectives 1. To ecome fmilir with the right endpoint rule, the trpezoidl rule, nd Simpson's rule. 2. To compre nd contrst the properties
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationMATH , Calculus 2, Fall 2018
MATH 362, 363 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationx = 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 information10 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 information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the xxis 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 informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationMath 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 information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Theorem Suppose f is continuous
More informationZ b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...
Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.
More information7.2 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 informationChapter 5. Numerical Integration
Chpter 5. Numericl Integrtion These re just summries of the lecture notes, nd few detils re included. Most of wht we include here is to be found in more detil in Anton. 5. Remrk. There re two topics with
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More informationMidpoint Approximation
Midpoint Approximtion Sometimes, we need to pproximte n integrl of the form R b f (x)dx nd we cnnot find n ntiderivtive in order to evlute the integrl. Also we my need to evlute R b f (x)dx where we do
More information( ) 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 information6.5 Numerical Approximations of Definite Integrals
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 6.5 Numericl Approximtions of Definite Integrls Sometimes the integrl of function cnnot be expressed with elementry functions, i.e., polynomil,
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationSection 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 informationUnit #10 De+inite Integration & The Fundamental Theorem Of Calculus
Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = x + 8x )Use
More informationCalculus 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 informationDistance And Velocity
Unit #8  The Integrl Some problems nd solutions selected or dpted from HughesHllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl
More information5.4. 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 informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2 elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More information