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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305270336

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Chapter
Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305270336
Chapter 7.7, Problem 20E
Textbook Problem
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(a) Find the approximations T10, and M10 for 1 2 e 1 / x d x .

(b) Estimate the errors in the approximations of part (a).

(c) How large do we have to choose n so that the approximations Tn and Mn to the integral in pan (a) are accurate to within 0.0001?

(a)

To determine

To Calculate: the approximate value of the integral using Trapezoidal rule and Midpoint rule.

Explanation of Solution

Given information:

The integral function is 12e1xdx.

The upper limit is 2 and lower limit is 1.

The number of sub intervals as n=10.

Calculation:

Calculate the approximate value of integral by using Trapezoidal rule.

Show the integral function as follows:

12e1xdx (1)

Show the Trapezoidal rule as follows:

abf(x)dxTn=Δx2[f(x0)+2f(x1)+2f(x2)+...+2f(xn1)+f(xn)]

Here, Tn is trapezoidal approximation, x1,x2,...xn are subintervals, Δx=ban and xi=a+iΔx.

Calculate the length of the subinterval (Δx) using the formula:

Δx=ban

Here, b is upper limit, a is lower limit, and n is number of subintervals.

Substitute 2 for b, 1 for a, and 10 for n.

Δx=2110=110=0.1

Hence, the subinterval is 0.1 for the limits [1,2].

The subinterval values are 1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 2 for x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, and x10.

Apply Trapezoidal rule in Equation (1).

12e1xdxTn=Δx2[f(x0)+2f(x1)+2f(x2)+...+2f(xn1)+f(xn)]

Substitute 0.1 for Δx and 10 for n.

T10=0.12[f(x0)+2f(x1)+2f(x2)+...+2f(x9)+f(x10)] (2)

Apply the subinterval values in Equation (2).

T10=0.12[f(x0)+2f(x1)+2f(x2)+...+2f(x9)+f(x10)]=0.12[f(1)+2f(1.1)+2f(1.2)+2f(1.3)+2f(1.4)+2f(1.5)+2f(1.6)+2f(1.7)+2f(1.8)+2f(1.9)+f(2)] (3)

Apply f(x)=e1x in Equation (3).

T10=0.12[f(1)+2f(1.1)+2f(1.2)+2f(1.3)+2f(1.4)+2f(1.5)+2f(1.6)+2f(1.7)+2f(1.8)+2f(1.9)+f(2)]=0.12[e11+(2×e11.1)+(2×e11.2)+(2×e11.3)+(2×e11.4)+(2×e11.5)+(2×e11.6)+(2×e11.7)+(2×e11.8)+(2×e11

(b)

To determine

To Estimate: The error value in Trapezoidal rule and Midpoint rule for evaluation of the integral 12e1xdx.

(c)

To determine

To Find: The larger value of n if guaranteed error value is within 0.0001 for Trapezoidal rule and Midpoint rule.

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Chapter 7 Solutions

Single Variable Calculus: Early Transcendentals
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Ch. 7.1 - Evaluate the integral. 11. t4lntdtCh. 7.1 - Evaluate the integral. 12. tan12ydyCh. 7.1 - Evaluate the integral. 13. tcsc2tdtCh. 7.1 - Evaluate the integral. 14. xcoshaxdxCh. 7.1 - Evaluate the integral. 15. (lnx)2dxCh. 7.1 - Evaluate the integral. 16. z10zdzCh. 7.1 - Evaluate the integral. 17. e2sin3dCh. 7.1 - Evaluate the integral. 18. ecos2dCh. 7.1 - Evaluate the integral. 19. z3ezdzCh. 7.1 - Evaluate the integral. 20. xtan2xdxCh. 7.1 - Evaluate the integral. 21. xe2x(1+2x)2dxCh. 7.1 - Evaluate the integral. 22. (arcsinx)2dxCh. 7.1 - Evaluate the integral. 23. 01/2xcosxdxCh. 7.1 - Evaluate the integral. 24. 01(x2+1)exdxCh. 7.1 - Evaluate the integral. 25. 02ysinhydyCh. 7.1 - Evaluate the integral. 26. 12w2lnwdwCh. 7.1 - Evaluate the integral. 27. 15lnRR2dRCh. 7.1 - Evaluate the integral. 28. 02t2sin2tdtCh. 7.1 - Evaluate the integral. 29. 0xsinxcosxdxCh. 7.1 - Evaluate the integral. 30. 13arctan(1/x)dxCh. 7.1 - Evaluate the integral. 31. 15MeMdMCh. 7.1 - Evaluate the integral. 32. 12(lnx)2x3dxCh. 7.1 - Evaluate the integral. 33. 0/3sinxln(cosx)dxCh. 7.1 - Evaluate the integral. 34. 01r34+r2drCh. 7.1 - Evaluate the integral. 35. 12x4(lnx)2dxCh. 7.1 - Evaluate the integral. 36. 0tessin(ts)dsCh. 7.1 - First make a substitution and then use integration...Ch. 7.1 - First make a substitution and then use integration...Ch. 7.1 - First make a substitution and then use integration...Ch. 7.1 - First make a substitution and then use integration...Ch. 7.1 - First make a substitution and then use integration...Ch. 7.1 - First make a substitution and then use integration...Ch. 7.1 - Evaluate the indefinite integral. Illustrate, and...Ch. 7.1 - Evaluate the indefinite integral. Illustrate, and...Ch. 7.1 - Evaluate the indefinite integral. Illustrate, and...Ch. 7.1 - Evaluate the indefinite integral. 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(b)...Ch. 7.2 - Evaluate the integral. 1. sin2xcos3xdxCh. 7.2 - Evaluate the integral. 2. sin3cos4dCh. 7.2 - Evaluate the integral. 3. 0/2sin7cos5dCh. 7.2 - Evaluate the integral. 4. 0/2sin5xdxCh. 7.2 - Evaluate the integral. 5. sin5(2t)cos2(2t)dtCh. 7.2 - Evaluate the integral. 6. tcos5(t2)dtCh. 7.2 - Evaluate the integral. 7. 0/2cos2dCh. 7.2 - Evaluate the integral. 8. 02sin2(13)dCh. 7.2 - Evaluate the integral. 9. 0cos4(2t)dtCh. 7.2 - Evaluate the integral. 10. 0sin2tcos4tdtCh. 7.2 - Evaluate the integral. 11. 0/2sin2xcos2xdxCh. 7.2 - Evaluate the integral. 12. 0/2(2sin)2dCh. 7.2 - Evaluate the integral. 13. cossin3dCh. 7.2 - Evaluate the integral. 14. sin2(1/t)t2dtCh. 7.2 - Evaluate the integral. 15. cotxcos2xdxCh. 7.2 - Evaluate the integral. 16. tan2xcos3xdxCh. 7.2 - Evaluate the integral. 17. sin2xsin2xdxCh. 7.2 - Evaluate the integral. 18. sinxcos(12x)dxCh. 7.2 - Evaluate the integral. 19. tsin2tdtCh. 7.2 - Evaluate the integral. 20. xsin3xdxCh. 7.2 - Evaluate the integral. 21. tanxsec3xdxCh. 7.2 - Evaluate the integral. 22. tan2sec4dCh. 7.2 - Evaluate the integral. 23. tan2xdxCh. 7.2 - Evaluate the integral. 24. (tan2x+tan4x)dxCh. 7.2 - Evaluate the integral. 25. tan4xsec6xdxCh. 7.2 - Evaluate the integral. 26. 0/4sec6tan6dCh. 7.2 - Evaluate the integral. 27. tan3xsecxdxCh. 7.2 - Evaluate the integral. 28. tan2xsec3xdxCh. 7.2 - Evaluate the integral. 29. tan3xsec6xdxCh. 7.2 - Evaluate the integral. 30. 0/4tan4tdtCh. 7.2 - Evaluate the integral. 31. tan5xdxCh. 7.2 - Evaluate the integral. 32. tan2xsecxdxCh. 7.2 - Evaluate the integral. 33. xsecxtanxdxCh. 7.2 - Evaluate the integral. 34. sincos3dCh. 7.2 - Evaluate the integral. 35. /6/2cot2xdxCh. 7.2 - Evaluate the integral. 36. /4/2cot3xdxCh. 7.2 - Evaluate the integral. 37. /4/2cot5csc3dCh. 7.2 - Evaluate the integral. 38. /4/2csc4cot4dCh. 7.2 - Evaluate the integral. 39. cscxdxCh. 7.2 - Evaluate the integral. 40. /6/3csc3xdxCh. 7.2 - Evaluate the integral. 41. sin8xcos5xdxCh. 7.2 - Evaluate the integral. 42. sin2sin6dCh. 7.2 - Evaluate the integral. 43. 0/2cos5tcos10tdtCh. 7.2 - Evaluate the integral. 44. sinxsec5xdxCh. 7.2 - Evaluate the integral. 45. 0/61+cos2xdxCh. 7.2 - Evaluate the integral. 46. 0/41cos4dCh. 7.2 - Evaluate the integral. 47. 1tan2xsec2xdxCh. 7.2 - Evaluate the integral. 48. dxcosx1Ch. 7.2 - Evaluate the integral. 49. xtan2xdxCh. 7.2 - If 0/4tan6xsecxdx=I, express the value of...Ch. 7.2 - Evaluate the indefinite integral. 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Evaluate the integral. 24. 01xx2dxCh. 7.3 - Evaluate the integral. 25. x23+2xx2dxCh. 7.3 - Evaluate the integral. 26. x2(3+4x4x2)3/2dxCh. 7.3 - Evaluate the integral. 27. x2+2xdxCh. 7.3 - Evaluate the integral. 28. x2+1(x22x+2)2dxCh. 7.3 - Evaluate the integral. 29. x1x4dxCh. 7.3 - Evaluate the integral. 30. 0/2cost1+sin2tdtCh. 7.3 - (a) Use trigonometric substitution to show that...Ch. 7.3 - Evaluate x2(x2+a2)3/2dx (a) by trigonometric...Ch. 7.3 - Find the average value of f(x)=x21/x, 1 x 7.Ch. 7.3 - Find the area of the region bounded by the...Ch. 7.3 - Prove the formula A=12r2 for the area of a sector...Ch. 7.3 - Evaluate the integral dxx4x22 Graph the integrand...Ch. 7.3 - Find the volume of the solid obtained by rotating...Ch. 7.3 - Find the volume of the solid obtained by rotating...Ch. 7.3 - (a) Use trigonometric substitution to verify that...Ch. 7.3 - The parabola y=12x2 divides the disk x2 + y2 8...Ch. 7.3 - A torus is generated by rotating the circle x2 +...Ch. 7.3 - A charged rod of length L produces an electric...Ch. 7.3 - Find the area of the crescent-shaped region...Ch. 7.3 - A water storage tank has the shape of a cylinder...Ch. 7.4 - Write out the form of the partial fraction...Ch. 7.4 - Write out the form of the partial fraction...Ch. 7.4 - Write out the form of the partial fraction...Ch. 7.4 - Write out the form of the partial fraction...Ch. 7.4 - Write out the form of the partial fraction...Ch. 7.4 - Write out the form of the partial fraction...Ch. 7.4 - Evaluate the integral. 7. x4x1dxCh. 7.4 - Evaluate the integral. 8. 3t2t+1dtCh. 7.4 - Evaluate the integral. 9. 5x+1(2x+1)(x1)dxCh. 7.4 - Evaluate the integral. 10. y(y+4)(2y1)dyCh. 7.4 - Evaluate the integral. 11. 0122x2+3x+1dxCh. 7.4 - Evaluate the integral. 12. 01x4x25x+6dxCh. 7.4 - Evaluate the integral. 13. axx2bxdxCh. 7.4 - Evaluate the integral. 14. 1(x+a)(x+b)dxCh. 7.4 - Evaluate the integral. 15. 10x34x+1x23x+2dxCh. 7.4 - Evaluate the integral. 16. 12x3+4x2+x1x3+x2dxCh. 7.4 - Evaluate the integral. 17. 124y27y12y(y+2)(y3)dyCh. 7.4 - Evaluate the integral. 18. 123x2+6x+2x2+3x+2dxCh. 7.4 - Evaluate the integral. 19. 01x2+x+1(x+1)2(x+2)dxCh. 7.4 - Evaluate the integral. 20. 23x(35x)(3x1)(x1)2dxCh. 7.4 - Evaluate the integral. 21. dt(t21)2Ch. 7.4 - Evaluate the integral. 22. x4+9x2+x+2x2+9dxCh. 7.4 - Evaluate the integral. 23. 10(x1)(x2+9)dxCh. 7.4 - Evaluate the integral. 24. x2x+6x3+3xdxCh. 7.4 - Evaluate the integral. 25. 4xx3+x2+x+1dxCh. 7.4 - Evaluate the integral. 26. x2+x+1(x2+1)2dxCh. 7.4 - Evaluate the integral. 27. x3+4x+3x4+5x2+4dxCh. 7.4 - Evaluate the integral. 28. x3+6x2x4+6x2dxCh. 7.4 - Evaluate the integral. 29. x+4x2+2x+5dxCh. 7.4 - Evaluate the integral. 30. x32x2+2x5x4+4x2+3dxCh. 7.4 - Evaluate the integral. 31. 1x31dxCh. 7.4 - Evaluate the integral. 32. 01xx2+4x+13dxCh. 7.4 - Evaluate the integral. 33. 01x3+2xx4+4x2+3dxCh. 7.4 - Evaluate the integral. 34. x5+x1x3+1dxCh. 7.4 - Evaluate the integral. 35. 5x4+7x2+x+2x(x2+1)2dxCh. 7.4 - Evaluate the integral. 36. x4+3x2+1x5+5x3+5xdxCh. 7.4 - Evaluate the integral. 37. x23x+7(x24x+6)2dxCh. 7.4 - Evaluate the integral. 38. x3+2x2+3x2(x2+2x+2)2dxCh. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Make a substitution to express the integrand as a...Ch. 7.4 - Use integration by parts, together with the...Ch. 7.4 - Use integration by parts, together with the...Ch. 7.4 - Use a graph of f(x) = 1/(x2 2x 3) to decide...Ch. 7.4 - Evaluate 1x2+kdx by considering several cases for...Ch. 7.4 - Evaluate the integral by completing the square and...Ch. 7.4 - Evaluate the integral by completing the square and...Ch. 7.4 - The German mathematician Karl Weierstrass...Ch. 7.4 - Use the substitution in Exercise 59 to transform...Ch. 7.4 - Use the substitution in Exercise 59 to transform...Ch. 7.4 - Use the substitution in Exercise 59 to transform...Ch. 7.4 - Use the substitution in Exercise 59 to transform...Ch. 7.4 - Find the area of the region under the given curve...Ch. 7.4 - Find the area of the region under the given curve...Ch. 7.4 - Find the volume of the resulting solid if the...Ch. 7.4 - One method of slowing the growth of an insect...Ch. 7.4 - Factor x4 +1 as a difference of squares by first...Ch. 7.4 - The rational number 227 has been used as an...Ch. 7.4 - (a) Use integration by parts to show that, for any...Ch. 7.4 - Suppose that F, G, and Q are polynomials and...Ch. 7.4 - If f is a quadratic function such that f(0) = 1...Ch. 7.4 - If a 0 and n is a positive integer, find the...Ch. 7.5 - Evaluate the integral. 1. cosx1sinxdxCh. 7.5 - Evaluate the integral. 2. 01(3x+1)2dxCh. 7.5 - Evaluate the integral. 3. 14ylnydyCh. 7.5 - Evaluate the integral. 4. sin3xcosxdxCh. 7.5 - Evaluate the integral. 5. tt4+2dtCh. 7.5 - Evaluate the integral. 6. 01x(2x+1)3dxCh. 7.5 - Evaluate the integral. 7. 11earctany1+y2dyCh. 7.5 - Evaluate the integral. 8. tsintcostdtCh. 7.5 - Evaluate the integral. 9. 24x+2x2+3x4dxCh. 7.5 - Evaluate the integral. 10. cos(1/x)x3dxCh. 7.5 - Evaluate the integral. 11. 1x3x21dxCh. 7.5 - Evaluate the integral. 12. 2x3x3+3xdxCh. 7.5 - Evaluate the integral. 13. sin5tcos4tdtCh. 7.5 - Evaluate the integral. 14. ln(1+x2)dxCh. 7.5 - Evaluate the integral. 15. xsecxtanxdxCh. 7.5 - Evaluate the integral. 16. 02/2x21x2dxCh. 7.5 - Evaluate the integral. 17. 0tcos2tdtCh. 7.5 - Evaluate the integral. 18. 14ettdtCh. 7.5 - Evaluate the integral. 19. ex+exdxCh. 7.5 - Evaluate the integral. 20. e2dxCh. 7.5 - Evaluate the integral. 21.arctanxdxCh. 7.5 - Evaluate the integral. 22. lnxx1+(lnx)2dxCh. 7.5 - Evaluate the integral. 23. 01(1+x)8dxCh. 7.5 - Evaluate the integral. 24. (1+tanx)2secxdxCh. 7.5 - Evaluate the integral. 25. 011+12t1+3tdtCh. 7.5 - Evaluate the integral. 26. 013x2+1x3+x2+x+1dxCh. 7.5 - Evaluate the integral. 27. dx1+exCh. 7.5 - Evaluate the integral. 28. sinatdtCh. 7.5 - Evaluate the integral. 29. ln(x+x21)dxCh. 7.5 - Evaluate the integral. 30. 12|ex1|dxCh. 7.5 - Evaluate the integral. 31. 1+x1xdxCh. 7.5 - Evaluate the integral. 32. 13e3/xx2dxCh. 7.5 - Evaluate the integral. 33. 32xx2dxCh. 7.5 - Evaluate the integral. 34. /4/21+4cotx4cotxdxCh. 7.5 - Evaluate the integral. 35. /2/2x1+cos2xdxCh. 7.5 - Evaluate the integral. 36. 1+sinx1+cosxdxCh. 7.5 - Evaluate the integral. 37. 0/4tan3sec2dCh. 7.5 - Evaluate the integral. 38. /6/3sincotsecdCh. 7.5 - Evaluate the integral. 39. sectansec2secdCh. 7.5 - Evaluate the integral. 40. 0sin6xcos3xdxCh. 7.5 - Evaluate the integral. 41. tan2dCh. 7.5 - Evaluate the integral. 42. tan1xx2dxCh. 7.5 - Evaluate the integral. 43. x1+x3dxCh. 7.5 - Evaluate the integral. 44. 1+exdxCh. 7.5 - Evaluate the integral. 45. x5ex3dxCh. 7.5 - Evaluate the integral. 46. (x1)exx2dxCh. 7.5 - Evaluate the integral. 47. x3(x1)4dxCh. 7.5 - Evaluate the integral. 48. 01x21x2dxCh. 7.5 - Evaluate the integral. 49. 1x4x+1dxCh. 7.5 - Evaluate the integral. 50. 1x24x+1dxCh. 7.5 - Evaluate the integral. 51. 1x4x2+1dxCh. 7.5 - Evaluate the integral. 52. dxx(x4+1)Ch. 7.5 - Evaluate the integral. 53. x2sinhmxdxCh. 7.5 - Evaluate the integral. 54. (x+sinx)2dxCh. 7.5 - Evaluate the integral. 55. dxx+xxCh. 7.5 - Evaluate the integral. 56. dxx+xxCh. 7.5 - Evaluate the integral. 57. x3x+cdxCh. 7.5 - Evaluate the integral. 58. xlnxx21dxCh. 7.5 - Evaluate the integral. 59. dxx416Ch. 7.5 - Evaluate the integral. 60. dxx24x21Ch. 7.5 - Evaluate the integral. 61. d1+cosCh. 7.5 - Evaluate the integral. 62. d1+cos2Ch. 7.5 - Evaluate the integral. 63. xexdxCh. 7.5 - Evaluate the integral. 64. 1x+1dxCh. 7.5 - Evaluate the integral. 65. sin2x1+cos4xdxCh. 7.5 - Evaluate the integral. 66. /4/3ln(tanx)sinxcosxdxCh. 7.5 - Evaluate the integral. 67. 1x+1+xdxCh. 7.5 - Evaluate the integral. 68. x2x6+3x3+2dxCh. 7.5 - Evaluate the integral. 69. 131+x2x2dxCh. 7.5 - Evaluate the integral. 70. 11+2exexdxCh. 7.5 - Evaluate the integral. 71. e2x1+exdxCh. 7.5 - Evaluate the integral. 72. ln(x+1)x2dxCh. 7.5 - Evaluate the integral. 73. x+arcsinx1x2dxCh. 7.5 - Evaluate the integral. 74. 4x+10x2xdxCh. 7.5 - Evaluate the integral. 75. dxxlnxxCh. 7.5 - Evaluate the integral. 76. xxx2+1dxCh. 7.5 - Evaluate the integral. 77. xex1+exdxCh. 7.5 - Evaluate the integral. 78. 1+sinx1sinxdxCh. 7.5 - Evaluate the integral. 79. xsin2xcosxdxCh. 7.5 - Evaluate the integral. 80. secxcos2xsinx+secxdxCh. 7.5 - Evaluate the integral. 81. 1sinxdxCh. 7.5 - Evaluate the integral. 82. sinxcosxsin4x+cos4xdxCh. 7.5 - The functions y=ex2 and y=x2ex2 don't have...Ch. 7.5 - We know that F(x)=0teet is a continuous function...Ch. 7.6 - Use the indicated entry in the Table of Integrals...Ch. 7.6 - Use the indicated entry in the Table of Integrals...Ch. 7.6 - Use the indicated entry in the Table of Integrals...Ch. 7.6 - Use the indicated entry in the Table of Integrals...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - Use the Table of Integrals on Reference Pages 610...Ch. 7.6 - The region under the curve y = sin2.x from 0 to ...Ch. 7.6 - Find the volume of the solid obtained when the...Ch. 7.6 - Verify Formula 53 in the Table of Integrals (a) by...Ch. 7.6 - Verify Formula 31 (a) by differentiation and (b)...Ch. 7.7 - Let l=04f(x)dx where f is the function whose graph...Ch. 7.7 - The left, right. Trapezoidal, and Midpoint Rule...Ch. 7.7 - Estimate 01cos(x2)dx using (a) the Trapezoidal...Ch. 7.7 - Draw the graph of f(x)=sin(12x2) in the viewing...Ch. 7.7 - Use (a) the Midpoint Rule and (b) Simpsons Rule to...Ch. 7.7 - Use (a) the Midpoint Rule and (b) Simpsons Rule to...Ch. 7.7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7.7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7.7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7.7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7.7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7.7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7.7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7.7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7.7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7.7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7.7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7.7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7.7 - (a) Find the approximations T8 and M8 for the...Ch. 7.7 - (a) Find the approximations T10, and M10 for...Ch. 7.7 - (a) Find the approximations T10, M10 and S10 for...Ch. 7.7 - How large should n be to guarantee that the...Ch. 7.7 - Find the approximations Tn, Mn, and Sn for n = 6...Ch. 7.7 - Find the approximations Tn, Mn, and Sn for n = 6...Ch. 7.7 - Estimate the area under the graph in the figure by...Ch. 7.7 - The widths (in meters) of a kidney-shaped swimming...Ch. 7.7 - (a) Use the Midpoint Rule and the given data to...Ch. 7.7 - (a) A table of values of a function g is given....Ch. 7.7 - A graph of the temperature in Boston on August 11,...Ch. 7.7 - A radar gun was used to record the speed of a...Ch. 7.7 - The graph of the acceleration a(t) of a car...Ch. 7.7 - Water leaked from a tank at a rate of r(t) liters...Ch. 7.7 - The table (supplied by San Diego Gas and Electric)...Ch. 7.7 - Shown is the graph of traffic on an Internet...Ch. 7.7 - Use Simpsons Rule with n = 8 to estimate the...Ch. 7.7 - The table shows values of a force function f(x),...Ch. 7.7 - The region bounded by the curve y=1/(1+ex), the...Ch. 7.7 - The figure shows a pendulum with length L that...Ch. 7.7 - The intensity of light with wavelength traveling...Ch. 7.7 - Use the Trapezoidal Rule with n = 10 to...Ch. 7.7 - Sketch the graph of a continuous function on [0,...Ch. 7.7 - Sketch the graph of a continuous function on [0,...Ch. 7.7 - If f is a positive function and f(x)0foraxb, show...Ch. 7.7 - Show that if f is a polynomial of degree 3 or...Ch. 7.7 - Show that 12(Tn+Mn)=T2n.Ch. 7.7 - Show that 13Tn+23Mn=S2n.Ch. 7.8 - Explain why each of the following integrals is...Ch. 7.8 - Which of the following integrals are improper?...Ch. 7.8 - Find the area under the curve y=1/x3 from x = 1 to...Ch. 7.8 - (a) Graph the functions f(x)=1/x1.1 and...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Determine whether each integral is convergent or...Ch. 7.8 - Sketch the region and find its area (if the area...Ch. 7.8 - Sketch the region and find its area (if the area...Ch. 7.8 - Sketch the region and find its area (if the area...Ch. 7.8 - Sketch the region and find its area (if the area...Ch. 7.8 - Sketch the region and find its area (if the area...Ch. 7.8 - Sketch the region and find its area (if the area...Ch. 7.8 - (a) If g(x) = (sin2x)/x2, use your calculator or...Ch. 7.8 - (a) If g(x)=1/(x1), use your calculator or...Ch. 7.8 - Use the Comparison Theorem to determine whether...Ch. 7.8 - Use the Comparison Theorem to determine whether...Ch. 7.8 - Use the Comparison Theorem to determine whether...Ch. 7.8 - Use the Comparison Theorem to determine whether...Ch. 7.8 - Use the Comparison Theorem to determine whether...Ch. 7.8 - Use the Comparison Theorem to determine whether...Ch. 7.8 - The integral 01x(1+x)dx is improper for two...Ch. 7.8 - Evaluate 21xx24dx by the same method as in...Ch. 7.8 - Find the values of p for which the integral...Ch. 7.8 - Find the values of p for which the integral...Ch. 7.8 - Find the values of p for which the integral...Ch. 7.8 - (a) Evaluate the integral 0xnexdx for n = 0, 1, 2,...Ch. 7.8 - (a) Show that xdx is divergent. (b) Show that...Ch. 7.8 - The average speed of molecules in an ideal gas is...Ch. 7.8 - We know from Example 1 that the region R = {(x, y)...Ch. 7.8 - Use the information and data in Exercise 5.4.33 to...Ch. 7.8 - Find the escape velocity v0 that is needed to...Ch. 7.8 - Astronomers use a technique called stellar...Ch. 7.8 - A manufacturer of lightbulbs wants to produce...Ch. 7.8 - As we saw in Section 6.5, a radioactive substance...Ch. 7.8 - In a study of the spread of illicit drug use from...Ch. 7.8 - Dialysis treatment removes urea and other waste...Ch. 7.8 - Determine how large the number a has to be so that...Ch. 7.8 - Estimate the numerical value of 0ex2dx by writing...Ch. 7.8 - If f(t) is continuous for t 0, the Laplace...Ch. 7.8 - Show that if 0 f(t) Meat for t 0, where M and a...Ch. 7.8 - Suppose that 0 f(t) Meat and 0 f(t) Keat for t...Ch. 7.8 - If f(x)dx is convergent and a and b are real...Ch. 7.8 - Show that 0x2ex2dx=120ex2dx.Ch. 7.8 - Show that 0ex2dx=01lnydy interpreting the...Ch. 7.8 - Find the value of the constant C for which the...Ch. 7.8 - Find the value of the constant C for which the...Ch. 7.8 - Suppose f is continuous on [0, ) and limxf(x)=1....Ch. 7.8 - Show that if a 1 and b a + 1, then the following...Ch. 7 - State the rule for integration by parts. In...Ch. 7 - How do you evaluate sinmxcosnxdx if m is odd? What...Ch. 7 - If the expression a2x2 occurs in an integral, what...Ch. 7 - What is the form of the partial fraction...Ch. 7 - State the rules for approximating the definite...Ch. 7 - Define the following improper integrals. (a)...Ch. 7 - Define the improper integral abf(x)dx for each of...Ch. 7 - State the Comparison Theorem for improper...Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Determine whether the statement is true or false....Ch. 7 - Evaluate the integral. 1. 12(x+1)2xdxCh. 7 - Evaluate the integral. 2. 12x(x+1)2dxCh. 7 - Evaluate the integral. 3. esinxsecxdxCh. 7 - Evaluate the integral. 4. 0/6tsin2tdtCh. 7 - Evaluate the integral. 5. dt2t2+3t+1Ch. 7 - Evaluate the integral. 6. 12x5lnxdxCh. 7 - Evaluate the integral. 7. 0/2sin3cos2dCh. 7 - Evaluate the integral. 8. dxex1Ch. 7 - Evaluate the integral. 9. sin(lnt)tdtCh. 7 - Evaluate the integral. 10. 01arctanx1+x2dxCh. 7 - Evaluate the integral. 11. 12x21xdxCh. 7 - Evaluate the integral. 12. e2x1+e4xdxCh. 7 - Evaluate the integral. 13. ex3dxCh. 7 - Evaluate the integral. 14. x2+2x+2dxCh. 7 - Evaluate the integral. 15. x1x2+2xdxCh. 7 - Evaluate the integral. 16. sec6tan2dCh. 7 - Evaluate the integral. 17. xcoshxdxCh. 7 - Evaluate the integral. 18. x2+8x3x3+3x2dxCh. 7 - Evaluate the integral. 19. x+19x2+6x+5dxCh. 7 - Evaluate the integral. 20. tan5sec3dCh. 7 - Evaluate the integral. 21. dxx24xCh. 7 - Evaluate the integral. 22. costdtCh. 7 - Evaluate the integral. 23. dxxx2+1Ch. 7 - Evaluate the integral. 24. excosxdxCh. 7 - Evaluate the integral. 25. 3x3x2+6x4(x2+1)(x2+2)dxCh. 7 - Evaluate the integral. 26. xsinxcosxdxCh. 7 - Evaluate the integral. 27. 0/2cos3xsin2xdxCh. 7 - Evaluate the integral 28. x3+1x31dxCh. 7 - Evaluate the integral 29. 33x1+|x|dxCh. 7 - Evaluate the integral 30. dxex1e2xCh. 7 - Evaluate the integral 31. 0ln10exex1ex+8dxCh. 7 - Evaluate the integral 32. 0/4xsinxcos3xdxCh. 7 - Evaluate the integral 33. x2(4x2)3/2dxCh. 7 - Evaluate the integral 34. (arcsinx)2dxCh. 7 - Evaluate the integral 35. 1x+x3/2dxCh. 7 - Evaluate the integral 36. 1tan1+tandCh. 7 - Evaluate the integral 37. (cosx+sinx)2cos2xdxCh. 7 - Evaluate the integral 38. 2xxdxCh. 7 - Evaluate the integral 39. 01/2xe2x(1+2x)2dxCh. 7 - Evaluate the integral 40. /4/3tansin2dCh. 7 - Evaluate the integral or show that it is...Ch. 7 - Evaluate the integral or show that it is...Ch. 7 - Evaluate the integral or show that it is...Ch. 7 - Evaluate the integral or show that it is...Ch. 7 - Evaluate the integral or show that it is...Ch. 7 - Evaluate the integral or show that it is...Ch. 7 - Evaluate the integral or show that it is...Ch. 7 - Evaluate the integral or show that it is...Ch. 7 - Evaluate the integral or show that it is...Ch. 7 - Evaluate the integral or show that it is...Ch. 7 - Evaluate the indefinite integral. Illustrate and...Ch. 7 - Evaluate the indefinite integral. Illustrate and...Ch. 7 - Graph the function f(x) = cos2x sin3x and use the...Ch. 7 - Use the Table of Integrals on the Reference Pages...Ch. 7 - Use the Table of Integrals on the Reference Pages...Ch. 7 - Use the Table of Integrals on the Reference Pages...Ch. 7 - Use the Table of Integrals on the Reference Pages...Ch. 7 - Verify Formula 33 in the Table of Integrals (a) by...Ch. 7 - Verify Formula 62 in the Table of Integrals.Ch. 7 - Is it possible to find a number n such that 0xndx...Ch. 7 - For what values of a is 0eaxcosxdx convergent?...Ch. 7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 7 - Estimate the errors involved in Exercise 63, parts...Ch. 7 - Use Simpsons Rule with n = 6 to estimate the area...Ch. 7 - The speedometer reading (v) on a car was observed...Ch. 7 - A population of honeybees increased at a rate of...Ch. 7 - Suppose you are asked to estimate the volume of a...Ch. 7 - Use the Comparison Theorem to determine whether...Ch. 7 - Find the area of the region bounded by the...Ch. 7 - Find the area bounded by the curves y = cos x and...Ch. 7 - Find the area of the region bounded by the curves...Ch. 7 - The region under the curve y=cos2x,0x/2, is...Ch. 7 - The region in Exercise 75 is rotated about the...Ch. 7 - If f is continuous on [0, ) and limxf(x)=0, show...Ch. 7 - We can extend our definition of average value of a...Ch. 7 - Use the substitution u = 1/x to show that...Ch. 7 - The magnitude of the repulsive force between two...Ch. 7 - Three mathematics students have ordered a 14-inch...Ch. 7 - Evaluate 1x7xdx The straightforward approach would...Ch. 7 - Evaluate 01(1x731x37)dx.Ch. 7 - The centers of two disks with radius 1 are one...Ch. 7 - A man initially standing at the point O walks...Ch. 7 - A function f is defined by f(x)=0costcos(xt)dt0x2...Ch. 7 - If n is a positive integer, prove that...Ch. 7 - Show that 01(1x2)ndx=22n(n!)2(2n+1)! Hint: Start...Ch. 7 - If 0 a b, find limt0{01[bx+a(1x)]tdx}1/tCh. 7 - Evaluate 1(x41+x6)2dx.Ch. 7 - Evaluate tanxdx.Ch. 7 - The circle with radius 1 shown in the figure...

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