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CALCULUS (INTEGRATION nd APPLICATIONS) THE ENTIRE COURSE IN ONE BOOK Grnt Skene for Grnt s Tutoring (www.grntstutoring.com) DO NOT RECOPY Grnt s Tutoring is privte tutoring orgniztion nd is in no wy ffilited with the University of Mnitob.

HOW TO USE THIS BOOK I hve broken the course up into lessons. Note tht ll the Lecture Problems for ll of the lessons re t the strt of this book (pges 6 to 9). Then I tech ech lesson, incorporting these Lecture Problems when pproprite. Mke sure you red my suggestion bout Index Crds on pge 5. If you re ble to solve ll the Prctise Problems I hve given you, then you should hve nothing to fer bout your Midterm or Finl Exm. I hve presented the course in wht I consider to be the most logicl order. Although my books re designed to follow the course syllbus, it is possible your prof will tech the course in different order or omit topic. It is lso possible he/she will introduce topic I do not cover. Mke sure you re ttending your clss regulrly! Sty current with the mteril, nd be wre of wht topics re on your exm. Never forget, it is your prof tht decides wht will be on the exm, so py ttention. If you hve ny questions or difficulties while studying this book, or if you believe you hve found mistke, do not hesitte to contct me. My phone number nd website re noted t the bottom of every pge in this book. Grnt s Tutoring is lso in the phone book. I welcome your input nd questions. Wishing you much success, Grnt Skene Owner of Grnt s Tutoring Grnt Skene for Grnt s Tutoring (text or cll (04) 489-884) DO NOT RECOPY Grnt s Tutoring is privte tutoring orgniztion nd is in no wy ffilited with the University of Mnitob.

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Four wys Grnt cn help you: Grnt s Study Books Bsic Sttistics (Stt 000) Bsic Sttistics (Stt 000) Liner Algebr nd Vector Geometry (Mth 300) Mtrices for Mngement (Mth 30) Intro Clculus (Mth 500 or Mth 50) Clculus for Mngement (Mth 50) Clculus (Mth 700 or 70) All these books re vilble t UMSU Digitl Copy Centre, room 8 University Centre, University of Mnitob. Grnt s books cn be purchsed there ll yer round. You cn lso order book from Grnt directly. Plese llow one business dy becuse the books re mde-to-order. Grnt s One-Dy Exm Prep Seminrs These re one-dy, -hour mrthons designed to explin nd review ll the key concepts in preprtion for n upcoming midterm or finl exm. Don t dely! Go to www.grntstutoring.com right now to see the dte of the next seminr. A seminr is generlly held one or two weeks before the exm, but don t risk missing it just becuse you didn t check the dte well in dvnce. You cn lso reserve your plce t the seminr online. You re not obligted to ttend if you reserve plce. You only py for the seminr if nd when you rrive. Grnt s Weekly Tutoring Groups This is for the student who wnts extr motivtion nd help keeping on top of things throughout the course. Agin, go to www.grntstutoring.com for more detils on when the groups re nd how they work. Grnt s Audio Lectures For less thn the cost of hours of one-on-one tutoring, you cn listen to over 40 hours of Grnt teching this book. Her Grnt work through exmples, nd offer tht extr bit of explntion beyond the written word. Go to www.grntstutoring.com for more detils.

TABLE OF CONTENTS Formuls to Memorize... Index Crds...5 Lecture Problems...6 Lecture... 0 Lesson : Inverse Trigonometric Functions... 0 Lesson : The Fundmentl Theorem of Clculus... 30 Lesson 3: Riemnn Sums... 43 Lesson 4: The Method of u Substitution... 56 Lesson 5: Are between Curves... 63 Lesson 6: Volumes... 74 Lesson 7: Integrls of Trigonometric Functions... 9 Lesson 8: Integrtion by Prts... 99 Lesson 9: Integrting by Trig Substitution... 07 Lesson 0: Integrting Rtionl Functions... 5 Lesson : L Hôpitl s Rule... 6 Lesson : Improper Integrls nd the Comprison Theorem... 33 Lesson 3: Arc Length nd Surfce Are... 47 Lesson 4: Prmetric Equtions... 57 Lesson 5: Polr Curves... 76 Midterm Exms for Clculus (36.70)... 89 Finl Exms for Clculus (36.70)... 05 Solutions to Midterm Exms for Clculus (36.70)... 9 Solutions to Finl Exms for Clculus (36.70)... 53 Grnt Skene for Grnt s Tutoring (Questions? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

Grnt Skene for Grnt s Tutoring (Comments? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

CALCULUS (INTEGRATION) FORMULAS TO MEMORIZE Elementry Integrls Stndrd Integrls Trigonometric Integrls. Kdu= K du= Ku. 3. 4... 3. n u du = u n + u e du = e Note: e.g. u n+ x e dx = e x 5x 5x e dx = e + C 5 u u du = ln The 3 Frction Integrls du du = ln u u = u du Note: e.g. = + u du u dx = ln x + b x + b dx = ln x 5 + C x 5 tn = sin u u. sin udu = cosu Note: sinx dx = cosx x dx = x + C 3 e.g. sin( 3 ) cos( 3 ). cosudu = sin u Note: cosx dx = sinx x dx = x + C 5 e.g. cos( 5 ) sin( 5 ) 3. tn udu = ln sec u 4. sec udu = ln sec u+ tn u 5. cot udu = ln sin u 6. csc udu = ln csc u cot u 7. 8. sec csc udu = tn u udu = cot u 9. sec utn udu = sec u 0. csc ucot udu = cscu Grnt Skene for Grnt s Tutoring (Questions? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

CALCULUS (INTEGRATION) Differentition Rules n n The Power Rule: ( x ) nx = The Product Rule: ( ) f g = f g+ f g T T B T B The Quotient Rule: = B B u = nu u n n Chin Rule Version of Power Rule: ( ) Derivtives of Trigonometric Functions: ( sin u) ( cosu) cosu u = ( tn u) = sec u u ( ) sin u u = ( cot u) = csc u u ( ) Derivtives of Exponentil nd Logrithmic Functions: u ( ) u e e u u u u = ( ) ln u sec u = sec utn u u cscu = cscucot u u = ( ln u) = ( log u ) = u ( u) Derivtives of Inverse Trigonometric Functions: u ln ( sin u) = u u = u u ( cos u) ( tn u) = u + u The Definition of the Definite Integrl: b n n b f( x) dx = lim f( x *) x = lim f + i P 0 n i= i= Summtion Formuls to Memorize: n ( b ) i n n i= = n n i= ( + ) n n i = n i= i n n = ( + )( n+ ) 6 u f t dt = f u u Fundmentl Theorem of Clculus: ( ) ( ) Grnt Skene for Grnt s Tutoring (Comments? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

CALCULUS (INTEGRATION) 3 Trigonometric Vlues to Memorize θ 0 sin θ 0 cos θ tn θ 0 π 6 3 3 π 4 π 3 3 π 0 3 undefined Trigonometric Identities to Memorize Pythgoren Identities: θ θ sin + cos = tn θ+ = sec cot θ+ = csc θ θ Hlf-Angle Identities: cos cos θ = ( + θ ) θ = ( θ ) sin cos sin θ = sin θcos θ Are between Two Curves: ( y y) dx or ( ) Volume of Solid of Revolution: x= b x= y= b y= x x dy If you wish to use dx nd re told to revolve bout the x-xis; or, if you wish to use dy nd re told to revolve bout the y-xis; then you will use the disk or wsher method: x= b Wsher Method: V = π ( y y ) dx or = π ( ) x= y= b V x x dy If you wish to use dx nd re told to revolve bout the y-xis; or, if you wish to use dy nd re told to revolve bout the x-xis; then you will use the cylindricl shell method: Cylindricl Shell Method: V = π x( y y) dx or = π ( ) x= b x= y= y= b V y x x dy y= Grnt Skene for Grnt s Tutoring (Questions? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

4 CALCULUS (INTEGRATION) b Averge Vlue of f(x) from to b: ( ) f x dx b Arc Length: b Let L = the length of curve from to b, then L = dl, where there re severl formuls for dl: Crtesin (x, y) curves: dl = dy + dx dx dx or dl = + dy dy Prmetric curves: dx dy dl = + dt dt dt Polr curves: dr dl = r + dθ dθ Surfce Are: Let S = the surfce re of curve from to b rotted bout the given xis, then: If the curve is rotted bout the x-xis: S = π y dl If the curve is rotted bout the y-xis: S = π x dl Are of Polr Curve: b b Let A = the re of polr curve from θ = to θ = b, then: A = b r dθ Grnt Skene for Grnt s Tutoring (Comments? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

CALCULUS (INTEGRATION) 5 INDEX CARDS To mke sure tht you lern to recognize wht technique of integrtion is required t the sight of n integrl, I suggest tht you mke 3 by 5 index crds for Lessons (question only), 4, 7, 8, 9 nd 0. As you re doing the homework for these lessons put ech question on seprte index crd. On the front of the crd put the integrl question, on the bck put note s to where the question cme from. Put just one question per crd nd, on the front, do not in nywy indicte where the question cme from (put tht informtion on the bck). Do this for the questions in the lesson nd the questions I hve suggested from the old exms. As you mss more nd more index crds you will be ble to shuffle them up. This wy, s you look t n integrl on the index crd, you won t know wht lesson it cme from. Thus, you will hve to be ble to identify the technique required to solve it just by looking t the integrl itself. BE SURE TO MAKE THESE CARDS. BELIEVE ME IT WORKS! Most students think they know the techniques, but don t relize tht the only reson they knew to use Integrtion by Prts, for exmple, to solve given question ws becuse they were doing the homework in the Integrtion by Prts lesson. Students who hve mde index crds hve become much more proficient t solving integrls without ny hint s to wht technique is required. Grnt Skene for Grnt s Tutoring (Questions? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

6 CALCULUS (INTEGRATION) Lecture Problems Lesson : Inverse Trigonometric Functions Trigonometric Vlues to Memorize π π π π θ 0 6 4 3 sin θ 0 cos θ tn θ 0 3 3 3 0 3 undefined Derivtives of Inverse Trigonometric Functions: ( sin u) = u u. Evlute the following: = u u ( cos u) ( tn u) = u + u () (c) sin 3 cos (e) tn ( 3) (g) (i) (b) (d) (f) sin cos 3 π sin sin 5. Compute dy dx () (c) y xsin x sin cos tn cos (h) cos( sin 0.7) (j) for the following (do not simplify): = (b) y = tn 3 5 x 3π tn tn 4 sin y = tn 3 ( x ) 5 ( x ) (d) y = cos ( x + x 3 ) Grnt Skene for Grnt s Tutoring (Comments? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

CALCULUS (INTEGRATION) 7 Homework for Lesson : Repet questions nd in this lesson. Do the following questions from the old exms: p. 89 #; p. 90 #; p.9, #3; p. 9 #; p. 93 #3; p. 94 # (b); p. 95 #9; p. 96 #5; p. 97 #5; p. 98 #; p. 99 #; p. 0 # (); p. 03 #4 (); p. #; p. 3 #; p. 4 #5 () nd (b). Note: the solutions to these questions begin on pge 9. Lesson : The Fundmentl Theorem of Clculus u f t dt = f u u ( ) ( ) b Averge Vlue of f(x) from to b: ( ) f x dx b Be sure to memorize ll the Elementry Integrls on pge.. Solve the following definite nd indefinite integrls: () cos( 3x) + 5 (c) π 4 x dx x 5 3 + dx x sin x (e) ( sin + sec 4 tn 4 ) 0 (b) x x x dx (f) 4 3 x + + x x ( ) (d) ( + ) 0 e x x dx 5x + csc 3 3 e x dx x dx 4 (g) f ( x) dx where f( x) 0 < = x + if x 3x if x 3 (h) 3 x 4 dx. Find f ( x) for the following functions: x dt + t () f( x) = 3 x f x = 5 3 (b) ( ) t (c) f ( x) = sin x e dt (d) f ( x) = x x3 y ( x) y ln cost dt 3 7 + 5 dy + 9 Grnt Skene for Grnt s Tutoring (Questions? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

8 CALCULUS (INTEGRATION) Homework for Lesson : While doing this homework mke index crds for question only (see pge 5 for n explntion). Repet questions nd in this lesson. Do the following questions from the old exms: p. 89 #; p.9, #; p. 93 #; p. 94 # (); p. 95 #6; p. 96 # nd #3; p. 98 # (e) nd #3; p. 99 # (e) nd #3; p. 00 # nd #5; p. 0 #3; p. 0 # (b); p. 03 #4 (b) nd (c); p. 05 # (); p. 09 #; p. 4 #9 Prt B; p. 6 #9; p. 7 #3. Note: the solutions to these questions begin on pge 9. Lesson 3: Riemnn Sums The Definition of the Definite Integrl: b n n b f( x) dx = lim f( x *) x = lim f + i P 0 n i= i= Summtion Formuls to Memorize: n ( b ) i n n i= = n n i= ( + ) n n i = n i= i n n = ( + )( n+ ) 6. Compute the left nd right Riemnn sums for the following functions over the given intervl using 4 equl prtitions. Include sketch of the region. () f( x) = 3x 4 on [0, 4] (b) ( ) f x = x 4x + 3 on [, 7]. Compute the lower nd upper Riemnn sums for the following functions over the given intervl using 4 equl prtitions. Include sketch of the region. () f( x) = 3x 4 on [0, 4] (b) ( ) f x = x 4x + 3 on [, 7] 3. Compute the Riemnn sum for the following functions over the given intervl using n equl prtitions. Then determine the limit s n pproches infinity ( n ). Check your nswer by computing the ssocited definite integrl. () f( x) = 3x 4 on [0, 4] (b) ( ) f x = x 4x + 3 on [, 7] Grnt Skene for Grnt s Tutoring (Comments? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

CALCULUS (INTEGRATION) 9 4. Solve the following limits: () n lim i n n n (b) = i n 3 3i lim + n i n n = 5 (c) n n π πi lim sin (d) lim n i = n n n i = n i + + n Homework for Lesson 3: Repet questions to 4 in this lesson. Do the following questions from the old exms: p. 94 #5; p. 97 #; p. 00 #; p. 0 #; p. 0 #4. Note: the solutions to these questions begin on pge 9. Lesson 4: The Method of u Substitution. Evlute the following: () 4 5x ( x ) 3 dx (b) 4x e 4 3 x dx (c) ln x + ln x dx x (d) x dx 6 + x (e) 5 3 x x 3 + dx (f) dx x + 6x + 0 (g) dx 4+ x x (h) x ( 3 e ) e x dx 3 (i) + dx (j) xdx x x x + 4x + 3 8 Grnt Skene for Grnt s Tutoring (Questions? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

0 CALCULUS (INTEGRATION) Homework for Lesson 4: While doing this homework mke index crds (see pge 5 for n explntion). Repet question in this lesson. Do the following questions from the old exms: p. 89 #3 nd #6; p. 90 # () nd (d); p.9, # (); p. 9 # () nd (b) nd #3; p. 93 # () nd (b); p. 94 # (); p. 95 #; p. 96 # (b); p. 97 #; p. 98 # () nd (f); p. 99 # () nd (b); p. 00 #3 (); p. 0 # () nd (b); p. 0 # (); p. 03 # () nd # (); p. 05 # (b) nd (c); p. 07 # (b); p. 09 #3 (); p. # (b); p. 3 #3 () nd (b); p. 4 #9 Prt A; p. 5 #3 (), (b) nd (d); p. # (); p. 3 # () nd (c); p. 5 # (); p. 7 # () nd (c). Note: the solutions to these questions begin on pge 9. Lesson 5: Are between Curves x= b ( y y) dx or ( ) x= y= b y= x x dy. Compute the res of the following bounded regions: () y 3 = x nd y 3 = x (b) x = y nd x = y y (c) (e) y = nd x + y = 5 (d) x 5x y = e, x = 0, x =, nd the x-xis y x y = nd = x + (f) consecutive intersections of y = sin x nd y = cos x. Let R be the region inside the circle centred t the origin of rdius 3 nd bove the line y = x 3. Set up BUT DO NOT EVALUATE the integrl (or integrls) tht would find the re of R. Homework for Lesson 5: Repet questions nd in this lesson. Do the following questions from the old exms: p. 90 #4 () nd (b) prt (i); p.9, #4; p. 9 #4 () nd (b); p. 93 #4 () nd (b); p. 95 #5; p. 96 #4; p. 98 #4; p. 99 #4; p. 00 #4; p. 0 #4; p. 0 #5; p. 03 #3; p. 09 #; p. #5 (); p. 3 # () nd (b); p. 5 # () nd (b); p. 7 #; p. #4 (); p. 8 #6 (). Note: the solutions to these questions begin on pge 9. Grnt Skene for Grnt s Tutoring (Comments? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

CALCULUS (INTEGRATION) Lesson 6: Volumes x= b Wsher Method: V = π ( y y ) dx or = π ( ) x= y= b V x x dy Cylindricl Shell Method: V = π x( y y) dx or = π ( ) x= b x= y= y= b V y x x dy. Compute the volumes of the following bounded regions, if the region is rotted bout: (i) the x-xis (ii) the y-xis y= () y 3 = x nd y = x (b) y x = nd x = 4 y y (c) y = sin x + nd y =, 0 x π (d) y = e, the x-xis, x = nd x = x. Given the region bounded by y = x nd bout the x-xis by using: y = x, find the volume if this region is rotted () the disk or wsher method (b) the cylindricl shell method. 3. For the region R bounded by following: = 4 nd y x x = 8, set up integrls for the y x x () The volume of the solid creted by rotting R bout the x-xis. (b) The volume of the solid creted by rotting R bout the y-xis. (c) The volume of the solid creted by rotting R bout the line x =. (d) The volume of the solid creted by rotting R bout the line y = 0. (e) The volume of the solid creted by rotting R bout the line x = 7. 4. Derive the formul for the volume of right circulr cone of height h nd bse rdius r by setting up nd solving volume integrl. 5. Derive the formul for the volume of sphere of rdius r by solving the pproprite volume integrl. 6. A solid hs circulr bse of rdius 3 units. Find the volume of the solid if prllel crosssections perpendiculr to the bse re squres. Grnt Skene for Grnt s Tutoring (Questions? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

CALCULUS (INTEGRATION) Homework for Lesson 6: Repet questions to 6 in this lesson. Do the following questions from the old exms: p. 90 #3 nd #4 (skip prt (i)); p.9, #5; p. 9 #4 (c) nd #5; p. 93 #4 (skip (b)); p. 94 #4 nd #5; p. 95 #7 nd #8; p. 96 #6; p. 97 #3; p. 06 #5 (c); p. 07 #4 (c) nd (d); p. 0 #6 (c); p. #5 (b); p. 3 # (c); p. 5 # (c); p. 7 #; p. 9 #3 nd #4; p. #4 (c); p. 4 #6 nd #7; p. 6 #5 () nd (b) nd #6; p. 7 #5; p. 8 #6 (b) nd (c). Note: the solutions to these questions begin on pge 9. Lesson 7: Integrls of Trigonometric Functions Trigonometric Identities to Memorize Pythgoren Identities: sin θ+ cos θ = tn θ+ = sec θ cot θ+ = csc θ Hlf-Angle Identities: cos cos θ = ( + θ ) θ = ( θ ) sin cos sin θ = sin θcos θ. Evlute the following: () (c) (e) cos x dx (b) 3 sin x dx (d) 3 sin x cos xdx (f) sin sin x cos x cos 3 6 tn x sec 4 xdx xdx xdx (g) cot x dx cos x (h) ( + ) cotx dx Homework for Lesson 7: While doing this homework mke index crds (see pge 5 for n explntion). Repet question in this lesson. Do the following questions from the old exms: p. 89 #4; p. 90 # (b); p.9, # (c); p. 9 # (d); p. 93 # (d); p. 94 # (b); p. 95 #; p. 96 # (); p. 97 #4 (b); p. 98 # (b); p. 00 #3 (b); p. 0 # (e); p. 0 #3 (); p. 03 # (b); p. 07 # (c); p. # (); p. 7 #4 (b); p. 9 # (d); p. 5 # (c). Note: the solutions to these questions begin on pge 9. Grnt Skene for Grnt s Tutoring (Comments? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

CALCULUS (INTEGRATION) 3 Lesson 8: Integrtion by Prts. Evlute the following: udv = uv vdu () x ln xdx (b) ( ln x ) dx (c) sin x dx (d) x x e dx 0 x dx (f) (e) x sin( 4 ) x sec xdx (g) e x cos3xdx (h) cos( ln x) dx Homework for Lesson 8: While doing this homework mke index crds (see pge 5 for n explntion). Repet question in this lesson. Do the following questions from the old exms: p. 89 #7; p. 90 # (c); p.9, # (b); p. 9 # (c); p. 93 # (c); p. 94 # (d); p. 95 #3; p. 96 # (d); p. 97 #4 (); p. 98 # (d); p. 99 # (d); p. 00 #3 (c); p. 0 # (c); p. 0 #3 (b); p. 03 # (b); p. 05 # (); p. 07 # (d); p. 09 #3 (c); p. 3 #3 (c) nd (d); p. 5 #3 (c); p. 7 #4 (); p. 9 # (b); p. # (e); p. 3 # (b); p. 5 # (b); p. 7 # (b). Note: the solutions to these questions begin on pge 9. Lesson 9: Integrting by Trig Substitution. Evlute the following: () x dx x (b) x dx 4 x (c) dx ( x 9) 3 + (d) x dx 4x 9 (e) dx ( 9x ) + (f) dx ( x x ) + + Grnt Skene for Grnt s Tutoring (Questions? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

4 CALCULUS (INTEGRATION) Homework for Lesson 9: While doing this homework mke index crds (see pge 5 for n explntion). Repet question in this lesson. Do the following questions from the old exms: p. 89 #5; p.9, # (d); p. 93 # (e); p. 94 # (c); p. 96 # (c); p. 98 # (c); p. 99 # (c); p. 00 #3 (d); p. 0 # (d); p. 0 # (b); p. 03 # (c); p. 05 # (c); p. # (c); p. 3 #3 (e); p. 5 #3 (e); p. 7 #5; p. 9 # (c); p. # (b); p. 3 # (d); p. 7 # (d). Note: the solutions to these questions begin on pge 9. Lesson 0: Integrting Rtionl Functions. Evlute the following: () x dx x + (b) x + 4x + 5 dx x x + (c) dx x 6x 7 (d) x + 3x + 4 ( x )( x )( x 3) dx x (e) dx 4 x (f) dx ( x )( x 4) + (g) dx ( x ) ( x 3) dx (h) 3 x x Homework for Lesson 0: While doing this homework mke index crds (see pge 5 for n explntion). Repet question in this lesson. Do the following questions from the old exms: p. 89 #8; p. 0 # (c); p. 05 # (b); p. 07 # (); p. 09 #3 (b); p. # (d); p. 3 #3 (f); p. 5 #3 (f); p. 7 #4 (c); p. 9 #; p. # (c); p. 3 #; p. 5 #; p. 7 #. Note: the solutions to these questions begin on pge 9. Grnt Skene for Grnt s Tutoring (Comments? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

CALCULUS (INTEGRATION) 5 Lesson : L Hôpitl s Rule. Solve the following limits: x sin x () lim x 0 3 x (b) x lim x 0 sin x (c) ( ) lim sec x cos3x ( ) x π (d) lim + x 0 x ln x (e) (g) 3 x lim x e x lim cos x x x x (f) lim ( x + e ) x 0 x (h) lim ( + sin 4x ) + x 0 cot x Homework for Lesson : Repet question in this lesson. Do the following questions from the old exms: p. 05 #3; p. 07 #; p. 09 #4; p. #3; p. 3 #4; p. 5 #4; p. 7 #6; p. 0 #6; p. #; p. 3 #3; p. 5 #3; p. 7 #4. Note: the solutions to these questions begin on pge 9. Lesson : Improper Integrls nd the Comprison Theorem. Determine if the following improper integrls converge or diverge. Evlute those tht re convergent. () 5 dx x (b) π 3 π 3 sec x dx dx (c) + x (d) 3 0 dx x Grnt Skene for Grnt s Tutoring (Questions? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

6 CALCULUS (INTEGRATION). Use the Comprison Test to determine if the following integrls converge or diverge. () 4 dx x + 0 x 3 (b) dx x + x 3 (c) sin x dx x (d) + x dx x (e) dx x x + e (f) 0 e x x dx Homework for Lesson : Repet questions nd in this lesson. Do the following questions from the old exms: p. 05 #4; p. 07 #3; p. 09 #5; p. #4; p. 4 #5; p. 6 #5; p. 8 #7; p. 0 #7; p. #3; p. 3 #4; p. 5 #4; p. 7 #3. Note: the solutions to these questions begin on pge 9. Lesson 3: Arc Length nd Surfce Are dy dl = + dx dx dx or dl = + dy If the curve is rotted bout the x-xis: S = π y dl If the curve is rotted bout the y-xis: S = π x dl dy b b. Find the lengths of the following curves: () y ex, 0 x = + (b) ( ) x = ln sin y, π 6 y π 3 (c) 3 x y = +, x (d) 6 x x ln x y =, x 4 4 Grnt Skene for Grnt s Tutoring (Comments? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

CALCULUS (INTEGRATION) 7. Set up but do not solve the integrls to find the surfce re if the curve on [0, ] is revolved bout: y x x 3 = + + 4 () the x-xis (c) the x = 3 line (b) the y-xis (d) the y = line 3. Find the surfce re obtined by rotting the first qudrnt region of the curve x = + y, x 5 bout: () the x-xis (solve this integrl) (b) the y-xis (set up the integrl, but do not solve it) Homework for Lesson 3: Repet questions to 3 in this lesson. Do the following questions from the old exms: p. 06 #5 () nd (b); p. 07 #4 (b) nd (e); p. 0 #6 () nd (b); p. #5 (c) nd (d); p. 4 #6; p. 6 #6; p. 8 #8; p. 0 #5; p. #4 (b); p. 4 #8; p. 6 #5 (c); p. 8 #7. Note: the solutions to these questions begin on pge 9. Lesson 4: Prmetric Equtions x= b dx dy dl = + dt dt dt ( ) or = ( ) Are = y y dx x= y= b Are x x dy y=. For the prmetric equtions given below find dy dx nd d y dx. () x = 3cos t, y = 5tnt (b) x = t + y = t t+ 3 4 5, 4 6 9. Find the eqution of the line tngent to the prmetric curve x = lnt, y t = te t t =. 3. Find the eqution of the line tngent to the prmetric curve x = t + 3, the point (5, 3). y t t = + t 4. Find the re bounded by the curve x = t t, y = t+ t nd the line y = 5/. Grnt Skene for Grnt s Tutoring (Questions? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

8 CALCULUS (INTEGRATION) 5. For the curve defined prmetriclly by x = t( t 3), y 3( t 3) () Show the point (0, 0) hs tngent lines. =, do the following: (b) Find the points where the curve hs either horizontl or verticl tngent line. (c) Sketch the curve. (d) Set up but do not solve the integrls expressing: (i) The circumference of the enclosed region. (ii) The re of the enclosed region. (iii) The surfce re of the solid creted by rotting the enclosed region bout the x-xis. 6. Sketch the prmetric curves defined below, by first doing the following: (i) Find the points which hve verticl or horizontl tngent lines. (ii) Use the first derivtive to estblish the directionl informtion of the curve. (iii) Use the second derivtive to estblish the concvity of the curve. () (b) x = t, t y = t 4t 3 x t, y t t = = [Hint: Verify tht this curve hs two tngents t (4, 0).] Homework for Lesson 4: Repet questions to 6 in this lesson. Do the following questions from the old exms: p. 06 #6; p. 08 #5; p. 0 #7; p. #6; p. 4 #7; p. 6 #7; p. 8 #9; p. 0 #8; p. #5; p. 4 #9; p. 6 #7; p. 8 #8. Note: the solutions to these questions begin on pge 9. Grnt Skene for Grnt s Tutoring (Comments? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

CALCULUS (INTEGRATION) 9 Lesson 5: Polr Curves dr dl = r + dθ dθ A = b r dθ. Sketch the following polr curves: () r = 5 (b) r = sinθ (c) r = 4cosθ (d) r = 3 ( cosθ) (e) r = + sinθ (f) r = sin θ (g) r = 4cos3θ (h) r = θ, θ. Find the re of one petl of the rose r = cosθ. 3. Set up but do not solve the integrl for the circumference of the crdioid r = + sinθ. 4. Find the slope of the tngent line to r = + cosθ t Homework for Lesson 5: π θ =. 6 Repet questions to 4 in this lesson. Do the following questions from the old exms: p. 0 #8; p. 4 #8; p. 6 #8; p. 8 #0; p. 0 #9; p. #6; p. 6 #8; p. 8 #9. Note: the solutions to these questions begin on pge 9. Grnt Skene for Grnt s Tutoring (Questions? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

0 CALCULUS (INTEGRATION) Grnt Skene for Grnt s Tutoring (Comments? Cll 489-884 or go to www.grntstutoring.com) DO NOT RECOPY

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Four wys Grnt cn help you: Grnt s Study Books Bsic Sttistics (Stt 000) Bsic Sttistics (Stt 000) Liner Algebr nd Vector Geometry (Mth 300) Mtrices for Mngement (Mth 30) Intro Clculus (Mth 500 or Mth 50) Clculus for Mngement (Mth 50) Clculus (Mth 700 or 70) All these books re vilble t UMSU Digitl Copy Centre, room 8 University Centre, University of Mnitob. Grnt s books cn be purchsed there ll yer round. You cn lso order book from Grnt directly. Plese llow one business dy becuse the books re mde-to-order. Grnt s One-Dy Exm Prep Seminrs These re one-dy, -hour mrthons designed to explin nd review ll the key concepts in preprtion for n upcoming midterm or finl exm. Don t dely! Go to www.grntstutoring.com right now to see the dte of the next seminr. A seminr is generlly held one or two weeks before the exm, but don t risk missing it just becuse you didn t check the dte well in dvnce. You cn lso reserve your plce t the seminr online. You re not obligted to ttend if you reserve plce. You only py for the seminr if nd when you rrive. Grnt s Weekly Tutoring Groups This is for the student who wnts extr motivtion nd help keeping on top of things throughout the course. Agin, go to www.grntstutoring.com for more detils on when the groups re nd how they work. Grnt s Audio Lectures For less thn the cost of hours of one-on-one tutoring, you cn listen to over 30 hours of Grnt teching this book. Her Grnt work through exmples, nd offer tht extr bit of explntion beyond the written word. Go to www.grntstutoring.com for more detils.