CALCULUS I. Practice Problems Integrals. Paul Dawkins

Transcription

1 CALCULUS I Practice Problems Integrals Paul Dawkins

2 Table of Contents Preface... Integrals... Introduction... Indefinite Integrals... Comuting Indefinite Integrals... Substitution Rule for Indefinite Integrals... 5 More Substitution Rule... Area Problem... 7 The Definition of the Definite Integral... 8 Comuting Definite Integrals... 9 Substitution Rule for Definite Integrals... Preface Here are a set of ractice roblems for my Calculus I notes. If you are viewing the df version of this document (as oosed to viewing it on the web) this document contains only the roblems themselves and no solutions are included in this document. Solutions can be found in a number of laces on the site.. If you d like a df document containing the solutions go to the note age for the section you d like solutions for and select the download solutions link from there. Or,. Go to the download age for the site htt://tutorial.math.lamar.edu/download.as and select the section you d like solutions for and a link will be rovided there.. If you d like to view the solutions on the web or solutions to an individual roblem you can go to the roblem set web age, select the roblem you want the solution for. At this oint I do not rovide df versions of individual solutions, but for a articular roblem you can select Printable View from the Solution Pane Otions to get a rintable version. Note that some sections will have more roblems than others and some will have more or less of a variety of roblems. Most sections should have a range of difficulty levels in the roblems although this will vary from section to section. Integrals 7 Paul Dawkins i htt://tutorial.math.lamar.edu/terms.as

3 Introduction Here are a set of ractice roblems for the Integrals chater of my Calculus I notes. If you are viewing the df version of this document (as oosed to viewing it on the web) this document contains only the roblems themselves and no solutions are included in this document. Solutions can be found in a number of laces on the site.. If you d like a df document containing the solutions go to the note age for the section you d like solutions for and select the download solutions link from there. Or, 5. Go to the download age for the site htt://tutorial.math.lamar.edu/download.as and select the section you d like solutions for and a link will be rovided there.. If you d like to view the solutions on the web or solutions to an individual roblem you can go to the roblem set web age, select the roblem you want the solution for. At this oint I do not rovide df versions of individual solutions, but for a articular roblem you can select Printable View from the Solution Pane Otions to get a rintable version. Note that some sections will have more roblems than others and some will have more or less of a variety of roblems. Most sections should have a range of difficulty levels in the roblems although this will vary from section to section. Here is a list of toics in this chater that have ractice roblems written for them. Indefinite Integrals Comuting Indefinite Integrals Substitution Rule for Indefinite Integrals More Substitution Rule Area Problem Definition of the Definite Integral Comuting Definite Integrals Substitution Rule for Definite Integrals Indefinite Integrals. Evaluate each of the following indefinite integrals. (a) (b) d 5 d Evaluate each of the following indefinite integrals. (a) d 7 Paul Dawkins htt://tutorial.math.lamar.edu/terms.as

4 (b) (c) + - 9d + + d- 9+ For roblems 5 evaluate the indefinite integral t t t dt w + w + wdw 9 7 z + z -z dz = Determine f ( ) given that f ( ) h t = t - t + t + t-. 7. Determine ht ( ) given that ( ) Comuting Indefinite Integrals For roblems evaluate the given integral d z - z - z dz t t t dt 5 w - w - 8dw dy. w+ w dw d 7 Paul Dawkins htt://tutorial.math.lamar.edu/terms.as

5 + - d y y y 9. dy. ( t - )( + ) t dt. Ê ˆ záz - dz Ë z. 8 5 z - z + z - dz z. - d. sin( ) + csc ( ) d 5. cos( )-sec( ) tan ( ) w w w dw. + ( ) È ( ) + ( ) csc q Îsin q csc q dq z dz e z t -t e - - -t e dt 9. - dw w w d cos. ( ) z + -z dz 7 Paul Dawkins htt://tutorial.math.lamar.edu/terms.as

6 . Determine f ( ) given that f ( ) = - and ( ) f - = 7. 7 z. Determine g( z ) given that g ( z) = z + -e and ( ) z g = 5 -e.. Determine ht ( ) given that h ( t) = t - 8t+, h ( ) =- 9 and ( ) h - =-. Substitution Rule for Indefinite Integrals For roblems evaluate the given integral.. ( 8 -)( - ) d ( + ) 7 t t dt. ( - )( - + 7). ( ) w w w dw 5 z- z -8zdz 5. 9 sin( + ) d. sec( -z) tan( - ) z dz ( 5-5 ) cos( + ) t t t t dt y -7y y y e dy 8. ( 7 - ) 9. w + dw w + w- ( cos - )( sin( ) - ) 5. ( ) t t t t dt 7 Paul Dawkins 5 htt://tutorial.math.lamar.edu/terms.as

7 Á Ë z Ê -zˆ -z. - e cos( e + ln ) e. sec ( v) + tan( v) dv z dz sin cos cos -5d. ( ) ( ) ( ). csc ( ) cot ( ) - ( ) csc 5. dy 7 + y d. -9w dw 7. Evaluate each of the following integrals. (a) d + 9 (b) ( + 9 ) (c) d + 9 d More Substitution Rule Evaluate each of the following integrals. 5+ 9t t dt. ( ) cos + -8 e d. ( ) 7 Paul Dawkins htt://tutorial.math.lamar.edu/terms.as

8 . 7w ( -8e ) 7w 7w e e + - 8e 7w cos( + ) 5. e dw d z ( z) ( z) sin cos 8 dz. e + e + 7w - -8w -8w wdw + 7t - 9t 5t + dt 7. ( ) 8. csc - Ê ˆ - Á d Ë 9. 7 ( + )( + ) + sin( + 8 ) y y y y dy. sec ( ) È9+ 7tan( )-tan ( ). 8 -w dw w + 9 t Î t t dt d 7. ( 8+ ) 8 z z dz Area Problem For roblems estimate the area of the region between the function and the -ais on the given interval using n= and using, (a) the right end oints of the subintervals for the height of the rectangles, (b) the left end oints of the subintervals for the height of the rectangles and, (c) the midoints of the subintervals for the height of the rectangles. 7 Paul Dawkins 7 htt://tutorial.math.lamar.edu/terms.as

9 . f ( ) = - + on [, ]. g( ) = - + on [-,]. h( ) =- cos( ) on [, ] 5. Estimate the net area between f ( ) = and the -ais on [,] - using n = 8 and the midoints of the subintervals for the height of the rectangles. Without looking at a grah of the function on the interval does it aear that more of the area is above or below the -ais? The Definition of the Definite Integral For roblems & use the definition of the definite integral to evaluate the integral. Use the right end oint of each interval for * i.. + d. ( - ) d. Evaluate : cos ( e + ) + d For roblems & 5 determine the value of the given integral given that ( ) ( ) g d =.. 9 f ( ) d 5. ( ) - ( ) g f d 9. Determine the value of f ( ) d 9 given that f ( ) d = and ( ) 5 5 f d =-7 and f d = 8. 7 Paul Dawkins 8 htt://tutorial.math.lamar.edu/terms.as

10 7. Determine the value of f ( ) d given that f ( ) d - =-, ( ) - ( ) f d =-. f d =9 and For roblems 8 & 9 sketch the grah of the integrand and use the area interretation of the definite integral to determine the value of the integral d 5 -d For roblems differentiate each of the following integrals with resect to.. 9cos ( - + ) t t dt. ( ) sin 7 t + dt. - t e - dt t Comuting Definite Integrals. Evaluate each of the following integrals. a. cos( ) - d 5 b. cos( ) - c. cos( ) - d 5 - d 5 Evaluate each of the following integrals, if ossible. If it is not ossible clearly elain why it is not ossible to evaluate the integral d 7 Paul Dawkins 9 htt://tutorial.math.lamar.edu/terms.as

11 z z dz w - w + wdw t - t dt. 7. z + - 7z z - + d - dz - 8. ( - ) - d 9. y - y dy y. 7sin() t - cos(). ( ) ( ) t dt sec z tan z - dz. sec ( ) 8csc( ) cot ( ) - w w wdw. e + d e y + dy y 5. f () t dt Ï t t > = Ì Ó - t t where f () t 7 Paul Dawkins htt://tutorial.math.lamar.edu/terms.as

12 . g ( z ) dz where g( z) - Ï- z z >- =Ì z Ó e z d w + dw - Substitution Rule for Definite Integrals Evaluate each of the following integrals, if ossible. If it is not ossible clearly elain why it is not ossible to evaluate the integral. 5. ( + )( + - ) d. 8cos ( t) ( t) 9-5sin dt. sin( z) cos ( ) z dz. - we w dw y + dy 5 - y. + e - d sin 7cos 7. ( w ) - ( ) 8. 5 w dw + - d Paul Dawkins htt://tutorial.math.lamar.edu/terms.as

13 9. t + t + dt - ( t-). ( - ) + sin( ) È+ cos( ) - z z Î z dz 7 Paul Dawkins htt://tutorial.math.lamar.edu/terms.as