Arnie Pizer Rochester Problem Library Fall 2005 WeBWorK assignment Integrals0Theory due 03/01/2006 at 01:00am EST.

Transcription

1 Arnie Pizer Rochester Problem Library Fall 5 WeBWorK assignment IntegralsTheory due 3//6 at :am EST.. ( pt) rochesterlibrary/setintegralstheory/osu in 4.pg You are given the four points in the plane A = (4,6), B = (6, 4), C = (,5), and D = (4, 4). The graph of the function f (x) consists of the three line segments AB, BC and 4 CD. Find the integral f (x)dx by interpreting the integral in 4 terms of sums and/or differences of areas of elementary figures. 4 f (x)dx = 4. ( pt) rochesterlibrary/setintegralstheory/sc5 4.pg Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry x dx 7 3. ( pt) rochesterlibrary/setintegralstheory/sc5 a.pg Use the Midpoint Rule to approximate 5.5 x 3 dx with n = ( pt) rochesterlibrary/setintegralstheory/sc5 8.pg Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry. 4 4x 7 dx 5. ( pt) rochesterlibrary/setintegralstheory/sc5 5.pg Use the Midpoint Rule to approximate the integral (x 3x )dx with n= ( pt) rochesterlibrary/setintegralstheory/sc5 3.pg Consider the integral 8 (4x + x + 4)dx (a) Find the Riemann sum for this integral using right endpoints and n = 3. R 3 = (b) Find the Riemann sum for this same integral, using left endpoints and n = 3. L 3 = 7. ( pt) rochesterlibrary/setintegralstheory/ur in.pg Consider the integral 8 4 ( ) x + dx. (a) Find the Riemann sum for this integral using right endpoints and n = 4. (b) Find the Riemann sum for this same integral, using left endpoints and n = ( pt) rochesterlibrary/setintegralstheory/ur in 3.pg Let f (x)dx =, f (x)dx =, f (x)dx = Find f (x)dx = and ( f (x) )dx = 8 9. ( pt) rochesterlibrary/setintegralstheory/sc5 3.pg 5 where a = and b = 8 b f (x) f (x) = f (x) a. ( pt) rochesterlibrary/setintegralstheory/ur in.pg Consider the function f (x) = x 7. In this problem you will calculate ) ( x 7 dx by using the definition b a 4 [ n f (x)dx = lim f (x i ) x n i= The summation inside the brackets is R n which is the Riemann sum where the sample points are chosen to be the righthand endpoints of each sub-interval. Calculate R n for f (x) = x 7 on the interval [,4] and write your answer as a function of n without any summation signs. You will need the summation formulas on page 38 of your textbook (page 364 in older texts). R n = lim R n = n ]

2 . ( pt) rochesterlibrary/setintegralstheory/osu in 5.pg The following sum 5 n + + n 5 n n n + 5n 5 n n + n 5 is a right Riemann sum for a certain definite integral b f (x)dx using a partition of the interval [,b] into n subintervals of equal length. Then the upper limit of integration must be: b = and the integrand must be the function f (x) =. ( pt) rochesterlibrary/setintegralstheory/osu in 6.pg The following sum ( ) 3 n ( ) 3 n n n ( ) 3 n n n is a right Riemann sum for the definite integral b f (x)dx where b = and f (x) = It is also a Riemann sum for the definite integral c g(x) dx 5 9 where c = and g(x) = The limit of these Riemann sums as n is 3. ( pt) rochesterlibrary/setintegralstheory/osu in 9.pg The following sum 4 ( ) n n + 4 ( ) 4 n n ( ) n n n is a right Riemann sum for the definite integral b f (x)dx where b = and f (x) = The limit of these Riemann sums as n is 4. ( pt) rochesterlibrary/setintegralstheory/osu in 7.pg Suppose f (x) is continuous and decreasing on the closed interval 6 x, that f (6) = 6, f () = and that Then 6 f (x)dx = 5. ( pt) rochesterlibrary/setintegralstheory/osu in 8.pg Consider the function f (x) = x 3 3x + 3x f (x)dx = By drawing a suitable picture, find a relation between the definite integrals f (x)dx and 64 find the second of these two integrals 64 f (x)dx = f (x)dx. Use this relation to 6. ( pt) rochesterlibrary/setintegralstheory/ur in.pg Estimate the area under the graph of f (x) = x + x from x = 5 to x = 8 using 3 approximating rectangles and left endpoints. 7. ( pt) rochesterlibrary/setintegralstheory/ur in.pg Evaluate the definite integral by interpreting it in terms of areas. 9 (x 6)dx 3 8. ( pt) rochesterlibrary/setintegralstheory/ur in 3.pg Given that 4 f (x) 5 for 7 x 5, use property 8 on page to estimate the value of f (x)dx 7 5 f (x)dx 7 9. ( pt) rochesterlibrary/setintegralstheory/csuf in.pg The interval [, 5] is partitioned into n equal subintervals, and a number x i is arbitrarily chosen in the i th subinterval for each i. Then n 6x i + 5 lim = n n i= Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

3 Arnie Pizer Rochester Problem Library Fall 5 WeBWorK assignment IntegralsInvTrig due 3//6 at :am EST.. ( pt) rochesterlibrary/setintegralsinvtrig/invtrigs.pg Evaluate the definite integral. sin( π 8 ) x 3 dx x. ( pt) rochesterlibrary/setintegralsinvtrig/invtrigs.pg Evaluate the definite integral. dx 34 + x 3. ( pt) rochesterlibrary/setintegralsinvtrig/invtrigs3.pg x 36 x dx 4. ( pt) rochesterlibrary/setintegralsinvtrig/invtrigs4.pg 5x 5 dx x +C 5. ( pt) rochesterlibrary/setintegralsinvtrig/invtrigs5.pg 4x x dx 6. ( pt) rochesterlibrary/setintegralsinvtrig/invtrigs6.pg 7 x 69x 6 dx 7. ( pt) rochesterlibrary/setintegralsinvtrig/invtrigs7.pg 6 dx 55 6x x 8. ( pt) rochesterlibrary/setintegralsinvtrig/osu in 6.pg dx = +C 5 8x WeBWorK notation for sin (x) is arcsin(x) or asin(x), for tan ( x) it s arctan(x) or atan(x). 9. ( pt) rochesterlibrary/setintegralsinvtrig/osu in 7.pg x dx = +C + x + 5 WeBWorK notation for sin (x) is arcsin(x) or asin(x), for tan ( x) it s arctan(x) or atan(x).. ( pt) rochesterlibrary/setintegralsinvtrig/ur in.pg For each of the indefinite integrals below, choose which of the following substitutions would be most helpful in evaluating the integral. Enter the appropriate letter (A,B, or C) in each blank. DO NOT EVALUATE THE INTEGRALS. A. x = 4tanθ B. x = 4sinθ C. x = 4secθ. (x 6) 5/ dx x dx. 6 x dx 3. (6 x ) 3/ 4. x 6dx 5. x 6 + x dx. ( pt) rochesterlibrary/setintegralsinvtrig/ur in.pg Match each of the trigonometric expressions below with the equivalent non-trigonometric function from the following list. Enter the appropriate letter (A,B,C,D, or E) in each blank. A. tan(arcsin(x/5)) B. cos(arcsin(x/5)) C. (/) sin( arcsin(x/5)) D. sin(arctan(x/5)) E. cos(arctan(x/5)) 5 x x x 3. 5 x 5 x x x 5. 5 x. ( pt) rochesterlibrary/setintegralsinvtrig/ur in 3.pg Evaluate the indefinite integral x 8 dx (49 x ) / 3. ( pt) rochesterlibrary/setintegralsinvtrig/ur in 4.pg Evaluate the definite integral 3 / 9 x dx 4. ( pt) rochesterlibrary/setintegralsinvtrig/ur in 5.pg Evaluate the indefinite integral dx (9 + x )

4 Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

5 Arnie Pizer Rochester Problem Library Fall 5 WeBWorK assignment IntegralsMethods due 3//6 at :am EST.. ( pt) rochesterlibrary/setintegralsmethods/ur in.pg x arctan(x)dx [NOTE: Remember to enter all necessary ( and )!! Enter arctan(x) for tan x, arcsin(x) for sin x. ]. ( pt) rochesterlibrary/setintegralsmethods/ur in.pg ln(x + x + 3)dx 3. ( pt) rochesterlibrary/setintegralsmethods/mec int.pg xcos (5x)dx 4. ( pt) rochesterlibrary/setintegralsmethods/mec int.pg (arcsinx) 7 dx x 5. ( pt) rochesterlibrary/setintegralsmethods/mec int3.pg e x e 4x + 64 dx 6. ( pt) rochesterlibrary/setintegralsmethods/osu in 3.pg Find the indicated integrals. ln(x 5 ) (a) dx = +C x e t cos(e t ) (b) 4 + 3sin(e t dt = +C ) 5/3 sin ( 3 (c) 5 x ) dx = 5 9x 7. ( pt) rochesterlibrary/setintegralsmethods/osu in 4.pg Find the indicated integrals (if they exist) x 4x + 6dx = e 6x e x + dx = 5x + 7 4x + 5x + 6 dx = ln(x) x 7 dx = Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

6 Arnie Pizer Rochester Problem Library Fall 5 WeBWorK assignment Integrals4Substitution due 3/4/6 at :am EST.. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5.pg Evaluate the integral x (x 3 ) 4 dx, by making the substitution u = x 3. NOTE: Your answer should be in terms of x and not u.. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4.pg Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral x 6 ( 4 + x 7) 8 dx Then the most appropriate substitution to simplify this integral is u = Then dx = f (x)du where f (x) = After making the substitution we obtain the integral g(u)du where g(u) = This last integral is: = +C (Leave out constant of integration from your answer.) After substituting back for u we obtain the following final form of the answer: = +C (Leave out constant of integration from your answer.) 3. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 4.pg Evaluate the integral by making the given substitution. dx, u = 8x + (8x + ) 3 4. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 6.pg Evaluate the integral by making the given substitution. sec(4x) tan(4x)dx, u = 4x 5. ( pt) rochesterlibrary/setintegrals4substitution/c4s5p.pg Evaluate the integral. x(x + 3) dx =. 6. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 7.pg (ln(x)) 4 dx x +C 7. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 9.pg 5 e x dx = + C 8. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 8.pg x 3 e x4 dx 9. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5.pg x 3 + x 3 dx. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 3.pg 4 (t + ) 5 dt. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5.pg cosx sinx + dx +C. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 4.pg x x + 6 dx [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan(x) for tan x, sin(x) for sinx. ] 3. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 5.pg 7 dx xln(x) 4. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 6.pg 5e 5x sin(e 5x )dx 5. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 9.pg x + x + x dx 6. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 9a.pg x + x + x + dx

7 7. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 3.pg x + 3 x + dx 8. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 3.pg 9x x 4 + dx 9. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 3a.pg 6x 5 (x 6 + ) dx =. ( pt) rochesterlibrary/setintegrals4substitution/sc pg x (x x + 6) 4 dx. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 33a.pg 4x (4x x + 4) 5 dx [NOTE: Remember to enter all necessary *, (, and )!!. ( pt) rochesterlibrary/setintegrals4substitution/sc pg 4sin 4 xcosxdx = 3. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4.pg Note: You can get full credit for this problem by just entering the answer to the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral cos (7t)sin(7t)dt Then the most appropriate substitution to simplify this integral is u = Then dt = f (t)du where f (t) = After making the substitution we obtain the integral g(u)du where g(u) = This last integral is: = +C (Leave out constant of integration from your answer.) After substituting back for u we obtain the following final form of the answer: = +C (Leave out constant of integration from your answer.) 4. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 5.pg Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the definite integral π/ π/6 cos(z) sin 4 (z) dz Then the most appropriate substitution to simplify this integral is u = Then dz = f (z)du where f (z) = After making the substitution and simplifying we obtain the integral b a g(u)du where g(u) = a = b = This definite integral has value = 5. ( pt) rochesterlibrary/setintegrals4substitution/ur in 4 3.pg sin 3 (8x)cos 5 (8x)dx +C 6. ( pt) rochesterlibrary/setintegrals4substitution/sc pg Evaluate the definite integral. π/3 e sin(x) cos(x)dx 7. ( pt) rochesterlibrary/setintegrals4substitution/sc pg Evaluate the definite integral. 3 + x dx 8. ( pt) rochesterlibrary/setintegrals4substitution/sc pg Evaluate the definite integral. π/ sin(t)dt 9. ( pt) rochesterlibrary/setintegrals4substitution/sc pg Evaluate the definite integral. 3 dx 4x ( pt) rochesterlibrary/setintegrals4substitution/sc5 5 5.pg Evaluate the definite integral. e 6 dx x lnx

8 3. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 7.pg Evaluate the definite integral. e 5 dx x( + lnx) 3. ( pt) rochesterlibrary/setintegrals4substitution/sc pg Verify that x = ( x ) x + and use this equation to evaluate 3 x dx 33. ( pt) rochesterlibrary/setintegrals4substitution/mec int3.pg e 5x e x + 9 dx 34. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4.pg 5 x dx = 35. ( pt) rochesterlibrary/setintegrals4substitution/sc5 5.pg Use the substitution x = 3 tan(θ) to evaluate the indefinite integral 38dx x x ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 4.pg Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral dx + (3x 8) Then the most appropriate substitution to simplify this integral is x = g(t) where g(t) = Note: We are using t as variable for angles instead of θ, since there is no standard way to type θ on a computer keyboard. After making this substitution and simplifying (using trig identities), we obtain the integral f (t)dt where f (t) = This integrates to the following function of t f (t)dt = +C After substituting back for t in terms of xwe obtain the following final form of the answer: +C 37. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 5.pg Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. /5 x 3 Consider the definite integral 5x dx / 5 Then the most appropriate substitution to simplify this integral is x = g(t) where g(t) = Note: We are using t as variable for angles instead of θ, since there is no standard way to type θ on a computer keyboard. After making this substitution and simplifying (using trig identities), we obtain the integral b f (t) = a = b = After evaluating this integral we obtain: /5 x 3 5x dx = / 5 a f (t)dt where 38. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 6.pg Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. 6 Consider the definite integral x 6x x dx Then the most appropriate substitution to simplify this integral is x = g(t) where g(t) = Note: We are using t as variable for angles instead of θ, since there is no standard way to type θ on a computer keyboard. After making this substitution and simplifying (using trig identities), we obtain the integral 8 b f (t) = a = b = After evaluating this integral we obtain: 6 x 6x x dx = 8 a f (t)dt where 39. ( pt) rochesterlibrary/setintegrals4substitution/mec int.pg (arcsinx) 4 dx x 3

9 4. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 3.pg Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral 5x + 8 x dx Then the most appropriate substitution to simplify this integral is u = Then dx = f (x)du where f (x) = After making the substitution and simplifying we obtain the integral g(u)du where g(u) = This last integral is: = +C (Leave out constant of integration from your answer.) After substituting back for u we obtain the following final form of the answer: = +C (Leave out constant of integration from your answer.) 4. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 4.pg Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Also the appropriate way to enter roots (except sqrt) into WeBWorK is to use fractional exponents. Consider the definite integral dx x x Then the most appropriate substitution to simplify this integral is u = Then dx = f (x)du where f (x) = After making the substitution and simplifying we obtain the integral b a g(u)du where g(u) = a = b = This definite integral has value = 4. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 6a.pg Calculate the following definite integral. x 9x + dx = +C 43. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 6.pg Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. 4 Consider the definite integral x 8x + 9dx Then the most appropriate substitution to simplify this integral is u = Then dx = f (x)du where f (x) = After making the substitution and simplifying we obtain the integral b a g(u)du where g(u) = a = b = This definite integral has value = 44. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 8a.pg Find the following indefinite integrals. x dx = +C x ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 8.pg Find the following indefinite integrals. x dx = +C x + 3 Hint: This is similar to Problem 6 of WeBWorK Hwwk #. cos(t) dt = +C (3sin(t) + 8) 46. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4.pg Note: You can get full credit for this problem by just entering the answer to the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral e x dx The most appropriate substitution to simplify this integral is u = f (x) where f (x) = We then have dx = g(u)du where g(u) = Hint: you need to back substitute for x in terms of u for this part. After substituting into the original integral we obtain h(u)du where h(u) = To evaluate this integral rewrite the numerator as 7 = u (u 7) simplify, then integrate, thus obtaining h(u)du = H(u) where H(u) = + C

10 After substituting back for u we obtain our final answer e x dx = + C 47. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 7.pg Consider the integral x x + 4x dx Then an appropriate trigonometric substitution to simplify this integral is x = f (t) where f (t) = After making this substitution and simplifying, we obtain the integral g(t) dt where g(t) = Note that this problem doesn t ask you to evaluate this integral. 48. ( pt) rochesterlibrary/setintegrals4substitution/osu in 4 8.pg For each of the following integrals find an appropriate trigonometric substitution of the form x = f (t) to simplify the integral. x = x = x = x = (5x 7) 3/ dx x 6x + 3 dx x 8x 3x + 6dx x 9 4x + 4x dx 49. ( pt) rochesterlibrary/setintegrals4substitution/csuf in 4.pg Evaluate the integral. 6 dx =. 4x + Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 5

11 Arnie Pizer Rochester Problem Library Fall 5 WeBWorK assignment Integrals5ByParts due 3/5/6 at :am EST.. ( pt) rochesterlibrary/setintegrals5byparts/sc5 6.pg Use integration by parts to evaluate the integral. xe 4x dx +C. ( pt) rochesterlibrary/setintegrals5byparts/osu in 5 5.pg x 4 e x dx = 3. ( pt) rochesterlibrary/setintegrals5byparts/sc5 6.pg Use integration by parts to evaluate the integral. 4x sin(x)dx 4. ( pt) rochesterlibrary/setintegrals5byparts/sc5 6 3.pg Use integration by parts to evaluate the integral. 5x cos(x)dx +C 5. ( pt) rochesterlibrary/setintegrals5byparts/sc5 6 4.pg Use integration by parts to evaluate the integral. 5xln(4x) dx +C 6. ( pt) rochesterlibrary/setintegrals5byparts/sc5 6 5.pg Use integration by parts to evaluate the integral. 5x cos(5x)dx +C 7. ( pt) rochesterlibrary/setintegrals5byparts/sc5 6 7.pg Use integration by parts to evaluate the integral. (ln(x)) dx +C 8. ( pt) rochesterlibrary/setintegrals5byparts/bennyparts.pg e 7x sin(7x)dx 9. ( pt) rochesterlibrary/setintegrals5byparts/sc5 6.pg Use integration by parts to evaluate the definite integral. e 6t lntdt. ( pt) rochesterlibrary/setintegrals5byparts/sc5 6 B.pg Evaluate the definite integral. 6 t 4 ln(3t)dt 3. ( pt) rochesterlibrary/setintegrals5byparts/sc5 6 5.pg Evaluate the definite integral. te t dt. ( pt) rochesterlibrary/setintegrals5byparts/sc5 6 6.pg Use integration by parts to evaluate the integral. 5 t lntdt 3. ( pt) rochesterlibrary/setintegrals5byparts/osu in 5 4.pg ytan (3y)dy = +C Use arctan() to denote tan () in your answer. 4. ( pt) rochesterlibrary/setintegrals5byparts/ur in 5.pg x arctan(7x)dx 5. ( pt) rochesterlibrary/setintegrals5byparts/mec int.pg xsin (8x)dx 6. ( pt) rochesterlibrary/setintegrals5byparts/sc5 6 6.pg First make a substitution and then use integration by parts to evaluate the integral. x 5 cos(x 3 )dx +C 7. ( pt) rochesterlibrary/setintegrals5byparts/ur in 5.pg ln(x + 9x + )dx 8. ( pt) rochesterlibrary/setintegrals5byparts/sc5 6 4.pg A particle that moves along a straight line has velocity v(t) = t e t meters per second after t seconds. How many meters will it travel during the first t seconds?

12 9. ( pt) rochesterlibrary/setintegrals5byparts/osu in 5 3.pg Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the definite integral /4 xsin (4x)dx The first step in evaluating this integral is to apply integration by parts: udv = uv vdu where u = and dv = h(x)dx where h(x) = Note: Use arcsin(x) for sin (x). /4 After integrating by parts, we obtain the integral f (x)dx on the right hand side where /4 vdu = f (x) = The most appropriate substitution to simplify this integral is x = g(t) where g(t) = Note: We are using t as variable for angles instead of θ, since there is no standard way to type θ on a computer keyboard. After making this substitution and simplifying (using trig identities), we obtain the integral b a k(t) dt where k(t) = a = b = After evaluating this integral and plugging back into the integration by parts formula we obtain: /4 xsin (4x)dx = Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

13 Arnie Pizer Rochester Problem Library Fall 5 WeBWorK assignment Integrals6Tables due 3/6/6 at :am EST.. ( pt) rochesterlibrary/setintegrals6tables/sc5 7 3.pg Use the Table of Integrals in the back of your textbook to evaluate the integral. e 6x sin5xdx. ( pt) rochesterlibrary/setintegrals6tables/tab int.pg Use the Table of Integrals in the back of your textbook to evaluate the integral. xdx (x + 4)ln(x + 4) 3. ( pt) rochesterlibrary/setintegrals6tables/tab int 5.pg Use the Table of Integrals in the back of your textbook to evaluate the integral. xdx x Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

14 Arnie Pizer Rochester Problem Library Fall 5 WeBWorK assignment Integrals7Approximations due 3/7/6 at :am EST.. ( pt) rochesterlibrary/setintegrals7approximations- /osu in 7.pg π/ Approximate sin(x)dx by computing L f (P) and U f (P), using the partition {, π/6, π/4, π/3, π/}. Your answers should be accurate to at least 4 decimal places. L f (P) = U f (P) = You may include a formula as an answer.. ( pt) rochesterlibrary/setintegrals7approximations- /osu in 7.pg π/ Approximate x sin(x)dx by computing L f (P) and U f (P), using the partition {, π/6, π/4, π/3, π/}. Your answers should be accurate to at least 4 decimal places. L f (P) = U f (P) = You may include a formula as an answer. 3. ( pt) rochesterlibrary/setintegrals7approximations- /osu in 7 3.pg Approximate the definite integral x dx using 4 subintervals of equal length and (a) midpoint rule (b) trapezoidal rule 7 (c) Simpson s rule 4. ( pt) rochesterlibrary/setintegrals7approximations- /osu in 7 4.pg Use the Midpoint Rule to approximate the integral (3x + x )dx with n= ( pt) rochesterlibrary/setintegrals7approximations/sc5 8 5.pg Given the following integral and value of n, approximate the integral using the methods indicated (round your answers to six decimal places): e 4x dx,n = 4 (a) Trapezoidal Rule (b) Midpoint Rule (c) Simpson s Rule 6. ( pt) rochesterlibrary/setintegrals7approximations/sc5 8 6.pg Use Simpson s Rule and all the data in the following table to 7 estimate the value of the integral ydx. x y Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

15 Arnie Pizer Rochester Problem Library Fall 5 WeBWorK assignment Integrals8Improper due 3/8/6 at :am EST.. ( pt) rochesterlibrary/setintegrals8improper/sc5 9 3.pg Find the area under the curve y = /(4x 3 ) from x = to x = t and evaluate it for t =,t =. Then find the total area under this curve for x. (a) t = (b) t = (c) Total area 6. ( pt) rochesterlibrary/setintegrals8improper/ur in 8.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. dx 7 x7/6. ( pt) rochesterlibrary/setintegrals8improper/sc5 9 3a.pg Find the area under the curve y = x 3 from x = 5 to x = t and evaluate it for t =, t =. Then find the total area under this curve for x 5. (a) t = (b) t = (c) Total area 3. ( pt) rochesterlibrary/setintegrals8improper/sc5 9 5.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent. 5e x dx 4. ( pt) rochesterlibrary/setintegrals8improper/ur in 8.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. e.3x dx. 7. ( pt) rochesterlibrary/setintegrals8improper/sc5 9 7.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent. 9 (x 3) dx 8. ( pt) rochesterlibrary/setintegrals8improper/sc5 9.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent. ((x + 6) + 6)dx 9. ( pt) rochesterlibrary/setintegrals8improper/sc5 9.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. x 7 e x8 dx. ( pt) rochesterlibrary/setintegrals8improper/ur in 8 3.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, give the answer -. xe 5x dx 5. ( pt) rochesterlibrary/setintegrals8improper/sc5 9 6.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent. 7 dx (x + 3) 3/. ( pt) rochesterlibrary/setintegrals8improper/sc5 9 6.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent. 5 x + dx

16 . ( pt) rochesterlibrary/setintegrals8improper/sc5 9 9.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent. ln(x) dx x 4 3. ( pt) rochesterlibrary/setintegrals8improper/ur in 8 4.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, give the answer -. ln(5x) dx x 5 4. ( pt) rochesterlibrary/setintegrals8improper/ur in 8 5.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. 6 dx x. 5. ( pt) rochesterlibrary/setintegrals8improper/sc5 9.pg Define the functions F(x) and G(x) by x x+5 F(x) = t 5 dt, G(x) = t 5 dt x x+5 Determine whether each of the following improper integrals and limits is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. (a) x 5 dx (b) (c) lim F(x) x lim G(x) x 6. ( pt) rochesterlibrary/setintegrals8improper/sc5 9.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. 9 (x 3) 3 dx 7. ( pt) rochesterlibrary/setintegrals8improper/sc5 9 6.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. 5 3 x 5 dx 8. ( pt) rochesterlibrary/setintegrals8improper/osu in 8 8.pg Consider the following integrals. Label each as P, C, D, according as the integral is proper, improper but convergent, or improper and divergent π/ π/ π 9π tan (7x)dx 3 dx x 9 x x + 9 dx se 7s ds t 49 dt ln(x 9)dx sin(7x) dx sin(x)tan (x)dx 9. ( pt) rochesterlibrary/setintegrals8improper/ur in 8 7.pg Let f (x) be a continuous function defined on the interval [, ) such that and Determine the value of f () = 8 f (x) < x f (x)e x/7 dx = f (x)e x/7 dx

17 Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 3

18 Arnie Pizer Rochester Problem Library Fall 5 WeBWorK assignment Integrals9Area due 3/9/6 at :am EST.. ( pt) rochesterlibrary/setintegrals9area/sc6 5.pg Sketch the region enclosed by y = 3x and y = 4x. Decide whether to integrate with respect to x or y. Then find the area of the region.. ( pt) rochesterlibrary/setintegrals9area/ns6.pg. ( pt) rochesterlibrary/setintegrals9area/sc6 7.pg Sketch the region enclosed by y = e 3x, y = e 8x, and x =. Decide whether to integrate with respect to x or y. Then find the area of the region. 3. ( pt) rochesterlibrary/setintegrals9area/sc6 9.pg Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. y = 3x,y = x ( pt) rochesterlibrary/setintegrals9area/sc6 9a.pg Find the area of the region enclosed by y = 5x and y = x ( pt) rochesterlibrary/setintegrals9area/sc6.pg Sketch the region enclosed by x+y = 3 and x+y =. Decide whether to integrate with respect to x or y. Then find the area of the region. 6. ( pt) rochesterlibrary/setintegrals9area/sc6 4.pg Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. y = 8cosx, y = (5sec(x)), x = π/4, x = π/4 7. ( pt) rochesterlibrary/setintegrals9area/ur in 9.pg Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. y = 3 x,y = 5, and y + 3x = 6 8. ( pt) rochesterlibrary/setintegrals9area/ns6 99.pg Find the area between the curves: y = x 3 x + x and y = x 3 + x x 9. ( pt) rochesterlibrary/setintegrals9area/osu in 9 5.pg The total area enclosed by the graphs of is y = x x 3 + x y = x + 9x Find the area enclosed between f (x) =.9x +3 and g(x) = x from x = 6 to x = 7.. ( pt) rochesterlibrary/setintegrals9area/ns6 5.pg Find the area of the region enclosed between y = 3sin(x) and y = cos(x) from x = to x =.6π. Hint: Notice that this region consists of two parts.. ( pt) rochesterlibrary/setintegrals9area/sc6 7.pg Use the parametric equations of an ellipse, x = cos(θ), y = 5sin(θ), θ π, to find the area that it encloses. 3. ( pt) rochesterlibrary/setintegrals9area/ns6 7.pg Use the parametric equations of an ellipse x = 9cosθ y = sinθ θ π to find the area that it encloses. 4. ( pt) rochesterlibrary/setintegrals9area/ur in 9.pg Find the area of the region enclosed by the parametric equation x = t 3 t y = 5t 5. ( pt) rochesterlibrary/setintegrals9area/ur in 9.pg There is a line through the origin that divides the region bounded by the parabola y = 7x 4x and the x-axis into two regions with equal area. What is the slope of that line?

19 6. ( pt) rochesterlibrary/setintegrals9area/ns6.pg Farmer Jones, and his wife, Dr. Jones, decide to build a fence in their field, to keep the sheep safe. Since Dr. Jones is a mathematician, she suggests building fences described by y = 3x and y = x +. Farmer Jones thinks this would be much harder than just building an enclosure with straight sides, but he wants to please his wife. What is the area of the enclosed region? 7. ( pt) rochesterlibrary/setintegrals9area/osu in 9 3.pg Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Find the area bounded by the two curves: x = ( y ) ( ) y x = 7 y The appropriate definite integral for computing this area has integrand ; lower limit of integration = ; and upper limit of integration = This definite integral has value = This is the area of the region enclosed by the two curves. 8. ( pt) rochesterlibrary/setintegrals9area/osu in 9 4.pg Consider the area between the graphs x +3y = 4 and x +6 = y. This area can be computed in two different ways using integrals First of all it can be computed as a sum of two integrals b a f (x)dx + c b g(x) dx where a =, b =, c = and f (x) = g(x) = Alternatively this area can be computed as a single integral β h(y) dy where α =, β = and h(y) = Either way we find that the area is. α 9. ( pt) rochesterlibrary/setintegrals9area/ur in 9.pg Find c > such that the area of the region enclosed by the parabolas y = x c and y = c x is 37. c = Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

20 Arnie Pizer Rochester Problem Library Fall 5 WeBWorK assignment IntegralsVolume due 3//6 at :am EST.. ( pt) rochesterlibrary/setintegralsvolume/osu in - /osu in.pg Consider the blue vertical line shown above (click on graph for better view) connecting the graphs y = g(x) = sin(x) and y = f (x) = cos(x). Referring to this blue line, match the statements below about rotating this line with the corresponding statements about the result obtained.. The result of rotating the line about the x-axis is. The result of rotating the line about the y-axis is 3. The result of rotating the line about the line y = is 4. The result of rotating the line about the line x = is 5. The result of rotating the line about the line x = π is 6. The result of rotating the line about the line y = is 7. The result of rotating the line about the line y = π 8. The result of rotating the line about the line y = π A. a cylinder of radius π x and height cos(x) sin(x) B. an annulus with inner radius + sin(x) and outer radius + cos(x) C. an annulus with inner radius π + sin(x) and outer radius π + cos(x) D. an annulus with inner radius π cos(x) and outer radius π sin(x) E. an annulus with inner radius cos(x) and outer radius sin(x) is F. an annulus with inner radius sin(x) and outer radius cos(x) G. a cylinder of radius x and height cos(x) sin(x) H. a cylinder of radius x + and height cos(x) sin(x). ( pt) rochesterlibrary/setintegralsvolume/osu in - /osu in.pg Consider the blue horizontal line shown above (click on graph for better view) connecting the graphs x = f (y) = sin(y) and x = g(y) = cos(3y). Referring to this blue line, match the statements below about rotating this line with the corresponding statements about the result obtained.. The result of rotating the line about the x-axis is. The result of rotating the line about the y-axis is 3. The result of rotating the line about the line y = is 4. The result of rotating the line about the line x = is 5. The result of rotating the line about the line x = π is 6. The result of rotating the line about the line y = is 7. The result of rotating the line about the line y = π 8. The result of rotating the line about the line y = π A. an annulus with inner radius + sin(y) and outer radius + cos(3y) B. a cylinder of radius y and height cos(3y) sin(y) C. a cylinder of radius π + y and height cos(3y) sin(y) D. a cylinder of radius π y and height cos(3y) sin(y) E. an annulus with inner radius π cos(3y) and outer radius π sin(y) is F. an annulus with inner radius sin(y) and outer radius cos(3y) G. a cylinder of radius y and height cos(3y) sin(y) H. a cylinder of radius + y and height cos(3y) sin(y) 3. ( pt) rochesterlibrary/setintegralsvolume/sc6.pg Find the volume of the solid obtained by rotating the region bounded by y = 6x, x =, and y =,, about the x-axis. V = 4. ( pt) rochesterlibrary/setintegralsvolume/sc6 3.pg Find the volume of the solid obtained by rotating the region bounded by y = x, y =, and x =,, about the y-axis. V =

21 5. ( pt) rochesterlibrary/setintegralsvolume/ns6.pg Find the volume formed by rotating the region enclosed by: x = 7y and y 3 = x with y about the y-axis 6. ( pt) rochesterlibrary/setintegralsvolume/ur in.pg Find the volume of the solid formed by rotating the region enclosed by y = e x + 3, y =, x =, x =.8 about the x-axis. 7. ( pt) rochesterlibrary/setintegralsvolume/ur in 3.pg Find the volume of the solid formed by rotating the region enclosed by y = e 3x +, y =, x =, x = about the y-axis. 8. ( pt) rochesterlibrary/setintegralsvolume/ur in 4.pg Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y = x and y = 5x about the x-axis. 9. ( pt) rochesterlibrary/setintegralsvolume/ur in 3.pg Find the volume of the solid formed by rotating the region enclosed by about the x-axis. x =, x =, y =, y = 4 + x 5. ( pt) rochesterlibrary/setintegralsvolume/ur in 4.pg Find the volume of the solid formed by rotating the region enclosed by 4. ( pt) rochesterlibrary/setintegralsvolume/sc6 9a.pg Find the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves x =, y =, x = y 3, about the line y =. 5. ( pt) rochesterlibrary/setintegralsvolume/ur in.pg Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = x,y = ; about y = 5 6. ( pt) rochesterlibrary/setintegralsvolume/sc6 4.pg Find the volume of the solid obtained by rotating the region bounded by y =, y =, x =, and x = 9, about y =. x5 V = 7. ( pt) rochesterlibrary/setintegralsvolume/ur in 5.pg Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y =,y = cos(7x),x = 4 π,x = about the axis y = 4 8. ( pt) rochesterlibrary/setintegralsvolume/osu in 9.pg The region between the graphs of y = x and y = 3x is rotated around the line y = 9. The volume of the resulting solid is 9. ( pt) rochesterlibrary/setintegralsvolume/osu in.pg The region between the graphs of y = x and y = 4x is rotated around the line x = 4. The volume of the resulting solid is. ( pt) rochesterlibrary/setintegralsvolume/osu in 3- /osu in 3.pg about the y-axis. x =, x =, y =, y = 3 + x 9. ( pt) rochesterlibrary/setintegralsvolume/sc6 4a.pg Find the volume of the solid obtained by rotating the region bounded by y =, y =, x = 3, and x = 9, about the y-axis. x4 V =. ( pt) rochesterlibrary/setintegralsvolume/sc pg Find the volume of the solid obtained by rotating the region bounded by y = 5x 3x and y = about the y-axis. V = 3. ( pt) rochesterlibrary/setintegralsvolume/sc6 9.pg Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = x 4,y = ; about y = 6 The base of a certain solid is the area bounded above by the graph of y = f (x) = 6 and below by the graph of y = g(x) = 9x. Cross-sections perpendicular to the x-axis are squares. (See picture above, click for a better view.) Use the formula b V = A(x) dx a to find the volume of the solid. Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. The lower limit of integration is a =

22 The upper limit of integration is b = The side s of the square cross-section is the following function of x: A(x)= Thus the volume of the solid is V =. ( pt) rochesterlibrary/setintegralsvolume/osu in 4- /osu in 4.pg The base of a certain solid is the area bounded above by the graph of y = f (x) = 36 and below by the graph of y = g(x) = 5x. Cross-sections perpendicular to the y-axis are squares. (See picture above, click for a better view.) Use the formula b V = A(y) dy a to find the volume of the formula. Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. The lower limit of integration is a = The upper limit of integration is b = The side s of the square cross-section is the following function of y: A(y)= Thus the volume of the solid is V =. ( pt) rochesterlibrary/setintegralsvolume/osu in 5- /osu in 5.pg applied to the picture shown above (click for a better view), with the left vertex of the triangle at the origin and the given altitude along the x-axis. Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. The lower limit of integration is a = The upper limit of integration is b = The diameter r of the semicircular cross-section is the following function of x: A(x)= Thus the volume of the solid is V = 3. ( pt) rochesterlibrary/setintegralsvolume/ns6 9.pg As a hardworking student, plagued by too much homework, you spend all night doing math homework. By 6am, you imagine yourself to be a region bounded by y = 5x x = x = 3 y = As you grow more and more tired, the world begins to spin around you. However, according to Newton, there is no difference between the world spinning around you, and you spinning around the world. Unfortunately, you are so tired that you think the world is the x-axis. What is the volume of the solid you (the region) create by spinning about the x-axis? 4. ( pt) rochesterlibrary/setintegralsvolume/ns6.pg You wake up one morning, and find yourself wearing a toga and scarab ring. Always a logical person, you conclude that you must have become an Egyptian pharoah. You decide to honor yourself with a pyramid of your own design. You decide it should have height h = 437 and a square base with side s = 3 To impress your Egyptian subjects, find the volume of the pyramid. 5. ( pt) rochesterlibrary/setintegralsvolume/ur in.pg A ball of radius 7 has a round hole of radius 9 drilled through its center. Find the volume of the resulting solid. 6. ( pt) rochesterlibrary/setintegralsvolume/ur in.pg Find the volume of a pyramid with height 5 and rectangular base with dimensions 6 and. The base of a certain solid is an equilateral triangle with altitude 3. Cross-sections perpendicular to the altitude are semicircles. Find the volume of the solid, using the formula b V = A(x) dx a 3 7. ( pt) rochesterlibrary/setintegralsvolume/mec.pg Find the volume of a pyramid with height and rectangular base with dimensions 6 and 9.

23 9. ( pt) rochesterlibrary/setintegralsvolume/osu in 8.pg The base of a certain solid is the triangle with vertices at (, 5), (5, 5), and the origin. Cross-sections perpendicular to the y-axis are squares. Then the volume of the solid is. 3. ( pt) rochesterlibrary/setintegralsvolume/osu in 6.pg A soda glass has the shape of the surface generated by revolving the graph of y = 7x for x about the y-axis. Soda is extracted from the glass through a straw at the rate of / cubic inch per second. How fast is the soda level in the glass dropping when the level is 4 inches? (Answer should be implicitly in units of inches per second. Do not put units in your answer. Also your answer should be positive, since we are asking for the rate at which the level DROPS rather than rises.) answer: 3. ( pt) rochesterlibrary/setintegralsvolume/osu in 7- /osu in 7.pg Coffee is poured into one of mugs above at a constant rate (constant volume per unit time). The graph below shows the depth of coffee in the mug as a function of time. (Click on images for better view.) Which mug was filled with coffee? Be prepared to explain your choice (offline). 3. ( pt) rochesterlibrary/setintegralsvolume/osu in.pg As viewed from above, a swimming pool has the shape of the ellipse x 6 + y 9 = The cross sections perpendicular to the ground and parallel to the y-axis are squares. Find the total volume of the pool. (Assume the units of length and area are feet and square feet respectively. Do not put units in your answer.) V = 33. ( pt) rochesterlibrary/setintegralsvolume/ur in 6.pg Find the volume of the solid obtained by rotating the region bounded by the curve y = sin(8x ) and the x-axis, x π8, about the y-axis. V = 34. ( pt) rochesterlibrary/setintegralsvolume/ur in 7.pg Find the volume of a right circular cone with height 6 and base radius 8. V = Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 4

24 Arnie Pizer Rochester Problem Library Fall 5 WeBWorK assignment IntegralsLength due 3//6 at :am EST.. ( pt) rochesterlibrary/setintegralslength/ur in.pg To find the length of the curve defined by y = 4x 3 + 7x from the point (,) to the point (4,84), you d have to compute b f (x)dx a where a =, b =, and f (x) =.. ( pt) rochesterlibrary/setintegralslength/ur in.pg Find the length of the curve defined by from x = 3 to x = 8. y = 3x 3/ 9 3. ( pt) rochesterlibrary/setintegralslength/ur in 3.pg Find the length of the curve defined by ( ( x ) ) y = ln from x = 4 to x = ( pt) rochesterlibrary/setintegralslength/ur in 4.pg Find the length of the arc formed by x = y 3 from point A to point B, where A = (,) and B = (4,) 5. ( pt) rochesterlibrary/setintegralslength/ur in 5.pg Find the length of the arc formed by y = ( 4x ln(x) ) 8 from x = 4 to x = 8 6. ( pt) rochesterlibrary/setintegralslength/osu in.pg Find the length of the curve L = x = 3y 4/3 3 3 y/3, 64 y 7 7. ( pt) rochesterlibrary/setintegralslength/ur in 6.pg Consider the parametric curve given by the equations x(t) = t + 3t + 3 y(t) = t + 3t 4 How many units of distance are covered by the point P(t) = (x(t),y(t)) between t=, and t=? 8. ( pt) rochesterlibrary/setintegralslength/ur in 3.pg Find the length of parametrized curve given by x(t) = t 3 8t 8t,y(t) = 4t 3 6t + 4t, where t goes from zero to one. Hint: The speed is a quadratic polynomial with integer coefficients. 9. ( pt) rochesterlibrary/setintegralslength/ur in.pg Let L be the circle in the x-y plane with center the origin and radius 95. Let S be a moveable circle with radius 7. S is rolled along the inside of L without slipping while L remains fixed. A point P is marked on S before S is rolled and the path of P is studied. The initial position of P is (95,). The initial position of the center of S is (5,). After S has moved counterclockwise about the origin through an angle t the position of P is ( ) 5 x = 5cost + 7cos 4 t ( ) 5 y = 5sint 7sin 4 t How far does P move before it returns to its initial position? Hint: You may use the formulas for cos( u+v) and sin( w /). S makes several complete revolutions about the origin before P returns to (95,).. ( pt) rochesterlibrary/setintegralslength/ns6 3 6.pg Consider the parametric equation x = 6(cosθ + θsinθ) y = 6(sinθ θcosθ) What is the length of the curve for θ = to θ = 7 4 π?. ( pt) rochesterlibrary/setintegralslength/ns6 3 8.pg If f (θ) is given by: f (θ) = cos 3 θ and g(θ) is given by: g(θ) = sin 3 θ Find the total length of the astroid described by f (θ) and g(θ). (The astroid is the curve swept out by ( f (θ),g(θ)) as θ ranges from to π. ). ( pt) rochesterlibrary/setintegralslength/ns6 3 5.pg Given the equation: xy =, set up an integral to find the length of path from x = a to x = b and enter the integrand below. (i.e. if your integral is L = R b a c x h dx enter c x h as your answer.) L = R b a dx