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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Repeated linear factors Evaluate the following integrals. 32. 122t3(t+1)dtRepeated linear factors Evaluate the following integrals. 33. x5x2(x+1)dxRepeated linear factors Evaluate the following integrals. 34. x2(x2)3dxIntegration Evaluate the following integrals. 47. x310x2+27xx210x+25dxIntegration Evaluate the following integrals. 48. x3+2x32x2+xdxRepeated linear factors Evaluate the following integrals. 37. x24x32x2+xdxIntegration Evaluate the following integrals. 50. 8(x2+4)x(x2+8)dxIntegration Evaluate the following integrals. 51. x2+x+2(x+1)(x2+1)dxIntegration Evaluate the following integrals. 52. x2+3x+2x(x2+2x+2)dxIntegration Evaluate the following integrals. 53. 2x2+5x+5(x+1)(x2+2x+2)dxIntegration Evaluate the following integrals. 54. z+1z(z2+4)dzIntegration Evaluate the following integrals. 55. 20x(x1)(x2+4x+5)dxIntegration Evaluate the following integrals. 56. 2x+1x2+4dxIntegration Evaluate the following integrals. 57. x3+5x(x2+3)2dxPreliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. 64. x4+1x3+9xdxIntegration Evaluate the following integrals. 59. x3+6x2+12x+6(x2+6x+10)2dxSimple irreducible quadratic factors Evaluate the following integrals. 50. dy(y2+1)(y2+2)Repeated quadratic factors Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. 83. 2x(x2+1)2dxRepeated quadratic factors Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. 84. dx(x+1)(x2+2x+2)2Integration Evaluate the following integrals. 63. 9x2+x+21(3x2+7)2dxIntegration Evaluate the following integrals. 64. 9x5+6x3(3x2+1)2dxExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. To evaluate 4x6x4+3x2dx, the first step is to find the partial fraction decomposition of the integrand. b. The easiest way to evaluate 6x+13x2+xdx is with a partial fraction decomposition of the integrand. c. The rational function f(x)=1x213x+42 has an irreducible quadratic denominator. a. The rational function f(x)=1x213x+43 has an irreducible quadratic denominator.66EAreas of regions Find the area of the following regions. 53. The region bounded by the curve y = 10/(x2 2x 24), the x-axis, and the lines x = 2 and x = 268EVolumes of solids Find the volume of the following solids. 57. The region bounded by y = x/(x + 1), the x-axis, and x = 4 is revolved about the x-axis.Volumes of solids Find the volume of the following solids.
The region bounded by y = y = 0, x = 1, and x = 2 is revolved about the y-axis.
Volumes of solids Find the volume of the following solids. 59. The region bounded by y=1x(3x), y = 0, x = 1, and x = 2 is revolved about the x-axis.72ETwo methods Evaluate dxx21, for x l, in two ways; using partial fractions and a trigonometric substitution. Reconcile your two answers.Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. 72. cos(sin34sin)dPreliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. 71. ex(ex1)(ex+2)dxPreliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. 74. dyy(ay), for a 0. (Hint: Let u=y.)Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. 69. sect1+sintdtPreliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. 70.ex+1dx(Hint: Let u=ex+1.)Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals. 81. (e3x+e2x+ex)(e2x+1)2dxPreliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals. 82. dxx1+2xPreliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. 63. dx1+exWhats wrong? Why are there no constants A and B satisfying x2(x4)(x+5)=Ax4+Bx+5?85E86ERational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u = tan (x/2) or equivalently x = 2 tan1 u. The following relations are used in making this change of variables. A:dx=21+u2duB:sinx=2u1+u2C:cosx=1u21+u2 88. Verify relation A by differentiating x = 2 tan1 u. Verify relations B and C using a right-triangle diagram and the double-angle formulas sinx=2sinx2cosx2andcosx=2cos2x21Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u = tan (x/2) or equivalently x = 2 tan1 u. The following relations are used in making this change of variables. A:dx=21+u2duB:sinx=2u1+u2C:cosx=1u21+u2 90. Evaluate dx2+cosx.Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u = tan (x/2) or equivalently x = 2 tan1 u. The following relations are used in making this change of variables. A:dx=21+u2duB:sinx=2u1+u2C:cosx=1u21+u2 91. Evaluate dx1cosx.90E91E92EThree start-ups Three cars. A, B, and C, start from rest and accelerate along a line according to the following velocity functions: vA(t)=88tt+1,vB(t)=88t2(t+1)2,andvC(t)=88t2t2+1. a. Which car travels farthest on the interval 0 t 1? b. Which car travels farthest on the interval 0 t 5? c. Find the position functions for each car assuming that each car starts at the origin. d. Which car ultimately gains the lead and remains in front?94E95E96EUse Table 8.1 (p. 520) to complete the process of evaluating sinx+1cos2xdx given in Example 1. Table 8.1 Basic Integration Formulas2QC3QCChoosing an integration strategy Identify a technique of integration for evaluating the following integrals if necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals. 1. 4xsin5xdx2EChoosing an integration strategy Identify a technique of integration for evaluating the following integrals if necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals. 3. x364x2dxChoosing an integration strategy Identify a technique of integration for evaluating the following integrals if necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals. 4. tan2x+1tanxdxChoosing an integration strategy Identify a technique of integration for evaluating the following integrals if necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals. 5. 5x2+18x+20(2x+3)(x2+4x+8)dx6EEvaluate the following integrals. 7. 0/2sin1+cosdEvaluate the following integrals. 8. cos210xdxEvaluate the following integrals. 9. 46dx8xx2Evaluate the following integrals. 10. sin9xcos3xdxEvaluate the following integrals. 11. 0/4(secxcosx)2dxEvaluate the following integrals. 12. ex1e2xdxEvaluate the following integrals. 13. dxex1e2xEvaluate the following integrals. 14. x2+x3x1+16x3dxEvaluate the following integrals. 15. 142xxdxEvaluate the following integrals. 16. dxx41Evaluate the following integrals. 17. 12w3ew2dwEvaluate the following integrals. 18. 5665x(x5)10dxEvaluate the following integrals. 19. 0/2sin7xdxEvaluate the following integrals. 20. 13dtt(t+1)Evaluate the following integrals. 21. x9ln3xdxEvaluate the following integrals. 22. dx(xa)(xb), a bEvaluate the following integrals. 23. sinxcos2x+cosxdxEvaluate the following integrals. 24. 3w5+2w412w312w32w34wdwEvaluate the following integrals. 25. dxx1x2Evaluate the following integrals. 26. 1/e1dxx(ln2x+2lnx+2)Evaluate the following integrals. 27. sin4x2dxEvaluate the following integrals. 28. 3x2+2x+3x4+2x2+1dxEvaluate the following integrals. 29. 2cosx+cotx1+sinxdxEvaluate the following integrals. 30. 5/253/2dvv225v2Evaluate the following integrals. 31. 369x2dx32EEvaluate the following integrals. 33. exa2+e2xdx, a 0Evaluate the following integrals. 34. 0/9sin3xcos3x+1dxEvaluate the following integrals. 35. 0/4(tan2+tan+1)sec2dEvaluate the following integrals. 36. x10xdxEvaluate the following integrals. 37. 0/6dx1sin2xEvaluate the following integrals. 38. /6/2cosxln(sinx)dxEvaluate the following integrals. 39. sinxln(sinx)dxEvaluate the following integrals. 40. sin2xln(sinx)dxEvaluate the following integrals. 41. cot3/2xcsc4xdxEvaluate the following integrals. 42. 01/2sin1x1x2dxEvaluate the following integrals. 43. x91x20dxEvaluate the following integrals. 44. dxx3x2Evaluate the following integrals. 45. 0ln21(1+ex)2dxEvaluate the following integrals. 46. dxe2x+1Evaluate the following integrals. 47. 2x3+x22x4x2x2dxEvaluate the following integrals. 48. 16x2x2dxEvaluate the following integrals. 49. tan3xsec9xdxEvaluate the following integrals. 50. tan7xsec4xdxEvaluate the following integrals. 51. 0/3tanxsec7/4xdxEvaluate the following integrals. 52. t2e3tdtEvaluate the following integrals. 53. excot3exdxEvaluate the following integrals. 54. 2x2+3x+26(x2)(x2+16)dxEvaluate the following integrals. 55. 3x2+3x+1x3+xdxEvaluate the following integrals. 56. 3/2sin2xesin2xdxEvaluate the following integrals. 57. sinxdxEvaluate the following integrals. 58. w2tan1wdwEvaluate the following integrals. 59. dxx4+x260EEvaluate the following integrals. 61. 02/2esin1xdx62EEvaluate the following integrals. 63. xalnxdx, a 164EEvaluate the following integrals. 65. 01/6dx19x266EEvaluate the following integrals. 67. x219x2dx68EEvaluate the following integrals. 69. dx1x2+1x270EEvaluate the following integrals. 71. 1cosx1+cosxdxEvaluate the following integrals. 72. x2sinhxdxEvaluate the following integrals. 73. 9161+xdxEvaluate the following integrals. 74. e3xex1dxEvaluate the following integrals. 75. 13tan1xx1/2+x3/2dxEvaluate the following integrals. 76. xx2+6x+18dxEvaluate the following integrals. 77. cos1xdx78EEvaluate the following integrals. 79. sin1xx2dxEvaluate the following integrals. 80. 214xx2dxEvaluate the following integrals. 81. x4+2x3+5x2+2x+1x5+2x3+xdxEvaluate the following integrals. 82. dx1+tanxEvaluate the following integrals. 83. exsin998(ex)cos3(ex)dxEvaluate the following integrals. 84. tan+tan3(1+tan)50dExplain why or why not Determine whether the following statements are true and give an explanation or counterexample.
More than one integration method can be used to evaluate .
Using the substitution in leads to .
The most efficient way to evaluate is to first rewrite the integrand in terms of sin 3x and cos 3x.
Using the substitution in leads to .
Area Find the area of the region bounded by the curves y=xx22x2, y=2x22x2 and x = 0.Surface area Find the area of the surface generated when the curve f(x) = sin x on [0, /2] is revolved about the x-axis.Volume Find the volume of the solid obtained by revolving the region bounded by the curve y=x1x2 on [0, 1] about the y-axisVolume Find the volume of the solid obtained by revolving the region bounded by the curve y=11sinx on [0, /4] about the x-axis.Work Let R be the region in the first quadrant bounded by the curve y=x44, and the lines y = 0 and y = 2. Suppose a tank that is full of water has the shape of a solid of revolution obtained by revolving region R about the y-axis. How much work is required to pump all the water to the top of the tank? Assume x and y are in meters.91E92E93EEvaluate the following integrals. 94. dtt3+195EEvaluate the following integrals. 96. ex3dx97E98E99EUse the result of Example 3 to evaluate 0/2dx1+sinx. Example 3 Using Tables of Integrals for Area Find the area of the region bounded by the curve y=11+sinx and the x-axis between x = 0 and x = .Using one computer algebra system, it was found that sinxcosxdx=12sin2x+C; using another computer algebra system, it was found that sinxcosxdx=12cos2x+C. Reconcile the two answers.3QCGive some examples of analytical methods for evaluating integrals.2E3EIs a reduction formula an analytical method or a numerical method? Explain.Evaluate excos3(ex)dx using tables after performing the substitution u = ex.Evaluate cosx100sin2xdx using tables after performing the substitution u = sin x.Table lookup integrals Use a table of integrals to determine the following indefinite integrals. 5. cos1xdxTable lookup integrals Use a table of integrals to determine the following indefinite integrals. 6. sin3xcos2xdxTable lookup integrals Use a table of integrals to determine the following indefinite integrals. 7. dxx2+16Table lookup integrals Use a table of integrals to determine the following indefinite integrals. 8. dxx225Table lookup integrals Use a table of integrals to determine the following indefinite integrals. 9. 3u2u+7duTable lookup integrals Use a table of integrals to determine the following indefinite integrals. 10. dyy(2y+9)Table lookup integrals Use a table of integrals to determine the following indefinite integrals. 11. dx1cos4xTable lookup integrals Use a table of integrals to determine the following indefinite integrals. 12. dxx81x2Table lookup integrals Use a table of integrals to determine the following indefinite integrals. 13. x4x+1dxTable lookup integrals Use a table of integrals to determine the following indefinite integrals. 14. t4t+12dtTable lookup integrals Use a table of integrals to determine the following indefinite integrals. 15. dx9x2100,x103Table lookup integrals Use a table of integrals to determine the following indefinite integrals. 16. dx22516x2Preliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 31. exe2x+4dxPreliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 32. ln2x+4xdxPreliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 33. cosxsin2x+2sinxdxPreliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 34. cos1xxdxPreliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 37. (lnx)sin1(lnx)xdxPreliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 38. dt1+4etTable lookup integrals Use a table of integrals to determine the following indefinite integrals. 17. dx(16+9x2)3/226ETable lookup integrals Use a table of integrals to determine the following indefinite integrals. 19. dxx144x2Table lookup integrals Use a table of integrals to determine the following indefinite integrals. 20. dvv(v2+8)Table lookup integrals Use a table of integrals to determine the following indefinite integrals. 21. ln2xdxTable lookup integrals Use a table of integrals to determine the following indefinite integrals. 22. x2e5xdxPreliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 23. x2+10xdx,x0Preliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 24. x28xdx,x8Preliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 25. dxx2+2x+1034EPreliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 27. dxx(x10+1)36EPreliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 29. dxx26x,x638EPreliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 35. tan1x3x4dxPreliminary work Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. 36. e3t4+e2tdtGeometry problems Use a table of integrals to solve the following problems. 39. Find the length of the curve y = x2/4 on the interval [0, 8].42E43EGeometry problems Use a table of integrals to solve the following problems. 42. The region bounded by the graph of y=x2lnx and the x-axis on the interval [1, e] is revolved about the x-axis. What is the volume of the solid that is formed?45EGeometry problems Use a table of integrals to solve the following problems. 44. Find the area of the region bounded by the graph of y=1x22x+2 and the x-axis between x = 0 and x = 3.47EGeometry problems Use a table of integrals to solve the following problems. 46. The graphs of f(x)=2x2+1 and g(x)=74x2+1 are shown in the figure. Which is greater, the average value of f or that of g on the interval [1, 1]?Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals. 67. x3e2xdxReduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals. 68. p2e3pdpReduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals. 69. tan43ydyReduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals. 70. sec44tdtDeriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is an integer. 79. xax+bdx (Use u = ax + b.)Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is an integer. 80. xax+bdx (Use u2 = ax + b.)Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is an integer. 81. x(ax+b)ndx(Use u = ax + b.)Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is an integer. 82. xnsin1xdx (Use integration by parts.)Apparent discrepancy Resolve the apparent discrepancy between dxx(x1)(x+2)=16ln(x1)2x+2x3+Canddxx(x1)(x+2)=lnx13+lnx+26lnx2+C.Evaluating an integral without the Fundamental Theorem of Calculus Evaluate 0/4ln(1+tanx)dx using the following steps. a. If f is integrable on [0, b], use substitution to show that 0bf(x)dx=0b/2(f(x)+f(bx))dx. b. Use part (a) and the identity tan(+)=tan+tan1tantan to evaluate 0/4ln(1+tanx)dx. (Source: The College Mathematics Journal 33, 4, Sep 2004)Two integration approaches Evaluate cos(lnx)dx two different ways: a. Use tables after first using the substitution u = ln x. b. Use integration by parts twice to verify your answer to part (a).Arc length of a parabola Let L(c) be the length of the parabola f(x) = x2 from x = 0 to x = c, where c 0 is a constant. a. Find an expression for L and graph the function. b. Is L concave up or concave down on [0, )? c. Show that as c becomes large and positive, the arc length function increases as c2; that is, L(c) kc2, where k is a constant.To apply the Midpoint Rule on the interval [3, 11] with n = 4, at what points must the integrand be evaluated?2QCCompute the approximate factor by which the error decreases in Table 8.5 between T(16) and T(32), and between T(32) and T(64).
4QC5QC6QCIf the interval [4, 18] is partitioned into n = 28 subintervals of equal length, what is x?Explain geometrically how the Midpoint Rule is used to approximate a definite integral.3EIf the Midpoint Rule is used on the interval [1, 11] with n = 3 subintervals, at what x-coordinates is the integrand evaluated?Compute the following estimates of 08f(x)dx using the graph in the figure. M(4)Compute the following estimates of 08f(x)dx using the graph in the figure. 6. T(4)7E8EIf the Trapezoid Rule is used on the interval [1, 9] with n = 5 subintervals, at what x-coordinates is the integrand evaluated?Suppose two Trapezoidal Rule approximations of abf(x)dx are T(2) = 6 and T(4) = 5.1. Find the Simpsons Rule approximation S(4).Absolute and relative error Compute the absolute and relative errors in using c to approximate x. 7. x = ; c = 3.14Absolute and relative error Compute the absolute and relative errors in using c to approximate x. 8. x=2; c = 1.414Midpoint Rule approximations Find the indicated Midpoint Rule approximations to the following integrals. 11. 2102x2dxusing n = 1, 2, and 4 subintervalsMidpoint Rule approximations Find the indicated Midpoint Rule approximations to the following integrals. 12. 19x3dx using n = 1, 2, and 4 subintervalsMidpoint Rule approximations Find the indicated Midpoint Rule approximations to the following integrals. 13.01sinxdxusing n = 6 subintervalsMidpoint Rule approximations Find the indicated Midpoint Rule approximations to the following integrals. 14. 01exdxusing n = 8 subintervalsTrapezoid Rule approximations Find the indicated Trapezoid Rule approximations to the following integrals. 15. 2102x2dxusing n = 2, 4, and 8 subintervals20ETrapezoid Rule approximations Find the indicated Trapezoid Rule approximations to the following integrals. 17. 01sinxdx using n = 6 subintervalsTrapezoid Rule approximations Find the indicated Trapezoid Rule approximations to the following integrals. 18. 01exdxusing n = 8 subintervalsSimpsons Rule approximations Find the indicated Simpsons Rule approximations to the following Integrals. 23. 0sinxdx using n = 4 and n = 6 subintervalsSimpsons Rule approximations Find the indicated Simpsons Rule approximations to the following Integrals. 24. 48xdx using n = 4 and n = 8 subintervalsSimpsons Rule approximations Find the indicated Simpsons Rule approximations to the following Integrals. 25. 23ex2dx using n = 10 subintervalsSimpsons Rule approximations Find the indicated Simpsons Rule approximations to the following Integrals. 26. 24cosxdxusing n = 8 subintervalsMidpoint Rule, Trapezoid Rule, and relative error Find the Midpoint and Trapezoid Rule approximations to 01sinxdx using n = 25 subintervals. Compute the relative error of each approximation.Midpoint Rule, Trapezoid Rule, and relative error Find the Midpoint and Trapezoid Rule approximations to 01exdx using n = 50 subintervals. Compute the relative error of each approximation.Comparing the Midpoint and Trapezoid Rules Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error. 21. 15(3x22x)dx=100Comparing the Midpoint and Trapezoid Rules Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error. 22. 26(x316x)dx=431E32E33EComparing the Midpoint and Trapezoid Rules Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error. 26. 08e2xdx=1e16235-36. River flow rates The following figure shows the discharge rate r(t) of the Snoqualmie River near Carnation, Washington, starting on a February day when the air temperature was rising. The variable t is the number of hours after midnight, r(t) is given in millions of cubic feet per hour, and dt equals the total amount of water that flows by the town of Carnation over a 24-hour period. Estimate dt using the Trapezoidal Rule and Simpson’s Rule with the following values of n.
35. n = 4
35-36. River flow rates The following figure shows the discharge rate r(t) of the Snoqualmie River near Carnation, Washington, starting on a February day when the air temperature was rising. The variable t is the number of hours after midnight, r(t) is given in millions of cubic feet per hour, and dt equals the total amount of water that flows by the town of Carnation over a 24-hour period. Estimate dt using the Trapezoidal Rule and Simpson’s Rule with the following values of n.
35. n = 6
Temperature data Hourly temperature data for Boulder, Colorado. San Francisco, California. Nantucket, Massachusetts, and Duluth, Minnesota, over a 12 hr period on the same day of January are shown in the figure. Assume that these data are taken from a continuous temperature function T(t). The average temperature over the 12-hr period isT=112012T(t)dt. 27. Find an accurate approximation to the average temperature over the 12-hr period for Boulder. State your method.Temperature data Hourly temperature data for Boulder, Colorado. San Francisco, California. Nantucket, Massachusetts, and Duluth, Minnesota, over a 12 hr period on the same day of January are shown in the figure. Assume that these data are taken from a continuous temperature function T(t). The average temperature over the 12-hr period isT=112012T(t)dt. 28. Find an accurate approximation to the average temperature over the 12-hr period for San Francisco. State your method.Temperature data Hourly temperature data for Boulder, Colorado. San Francisco, California. Nantucket, Massachusetts, and Duluth, Minnesota, over a 12 hr period on the same day of January are shown in the figure. Assume that these data are taken from a continuous temperature function T(t). The average temperature over the 12-hr period isT=112012T(t)dt. 29. Find an accurate approximation to the average temperature over the 12-hr period for Nantucket. State your method.Temperature data Hourly temperature data for Boulder, Colorado. San Francisco, California. Nantucket, Massachusetts, and Duluth, Minnesota, over a 12 hr period on the same day of January are shown in the figure. Assume that these data are taken from a continuous temperature function T(t). The average temperature over the 12-hr period isT=112012T(t)dt. 30. Find an accurate approximation to the average temperature over the 12-hr period for Duluth. State your method.Nonuniform grids Use the indicated methods to solve the following problems with nonuniform grids. 31. A curling iron is plugged into an outlet at time t = 0. Its temperature T in degrees Fahrenheit, assumed to be a continuous function that is strictly increasing and concave down on 0 t 120, is given at various times (in seconds) in the table. a. Approximate 11200120T(t)dt in three ways: using a left Riemann sum, a right Riemann sum, and the Trapezoid Rule. Interpret the value of 11200120T(t)dt in the context of this problem. b. Which of the estimates made in part (a) overestimates the value of 11200120T(t)dt? Underestimates? Justify your answers with a simple sketch of the sums you computed. c. Evaluate and interpret 0120T(t)dt in the context of this problem.Nonuniform grids Use ne indicated methods to solve the following problems with nonuniform grids. Approximating integrals The function f is twice differentiable on (, ). Values of f at various points on [0, 20] are given in the table. a. Approximate 0120f(x)dx in three ways: using a left Riemann sum, a right Riemann sum, and the Trapezoid Rule. b. A scatterplot of the data in the table is provided in the figure. Use the scatterplot to illustrate each of the approximations in part (a) by sketching appropriate rectangles for the Riemann sums and by sketching trapezoids for the Trapezoid Rule approximation. c. Evaluate 412(3f(x)+2)dx.Nonuniform grids Use the indicated methods to solve the following problems with nonuniform grids. 33. A hot-air balloon is launched from an elevation of 5400 ft above sea level. As it rises, its vertical velocity is computed using a device (called a variometer) that measures the change in atmospheric pressure. The vertical velocities at selected limes are shown in the table (with units of ft/min). a. Use the Trapezoid Rule to estimate the elevation of the balloon after five minutes. Remember that the balloon starts at an elevation of 5400 ft. b. Use a right Riemann sum to estimate the elevation of the balloon after five minutes. c. A polynomial that fits the data reasonably well is g(t)=3.49t343.21t2+142.43t1.75. Estimate the elevation of the balloon after five minutes using this polynomial.Nonuniform grids Use the indicated methods to solve the following problems with nonuniform grids. 34. A piece of wood paneling must be cut in the shape shown in the figure. The coordinates of several points on its curved surface are also shown (with units of inches) . a. Estimate the surface area of the paneling using the Trapezoid Rule. b. Estimate the surface area of the paneling using a left Riemann sum. c. Could two identical pieces be cut from a 9-in by 9-in piece of wood? Answer carefully.Trapezoid Rule and Simpsons Rule Consider the following integrals and the given values of n. a. Find the Trapezoid Rule approximations to the integral using n and 2n subintervals. b. Find the Simpsons Rule approximation to the integral using 2n subintervals. It is easiest to obtain Simpsons Rule approximations from the Trapezoid Rule approximations, as in Example 7. c. Compute the absolute errors in the Trapezoid Rule and Simpsons Rule with 2n subintervals. 35. 01e2xdx; n =25Trapezoid Rule and Simpsons Rule Consider the following integrals and the given values of n. a. Find the Trapezoid Rule approximations to the integral using n and 2n subintervals. b. Find the Simpsons Rule approximation to the integral using 2n subintervals. It is easiest to obtain Simpsons Rule approximations from the Trapezoid Rule approximations, as in Example 7. c. Compute the absolute errors in the Trapezoid Rule and Simpsons Rule with 2n subintervals. 36. 02x4dx; n = 30Trapezoid Rule and Simpsons Rule Consider the following integrals and the given values of n. a. Find the Trapezoid Rule approximations to the integral using n and 2n subintervals. b. Find the Simpsons Rule approximation to the integral using 2n subintervals. It is easiest to obtain Simpsons Rule approximations from the Trapezoid Rule approximations, as in Example 7. c. Compute the absolute errors in the Trapezoid Rule and Simpsons Rule with 2n subintervals. 37. 1edxx; n = 5048ESimpsons Rule Apply Simpsons Rule to the following integrals. It is easiest to obtain the Simpsons Rule approximations from the Trapezoid Rule approximations, as in Example 7. Make a table similar to Table 7.8 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error. 39. 04(3x58x3)dx=153650ESimpsons Rule Apply Simpsons Rule to the following integrals. It is easiest to obtain the Simpsons Rule approximations from the Trapezoid Rule approximations, as in Example 7. Make a table similar to Table 7.8 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error. 41. 0etsintdt=12(e+1)52EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Suppose abf(x)dx is approximated with Simpsons Rule using n = 18 subintervals, where |f(4)(x)| 1 on [a, b]. The absolute error ES in approximating the integral satisfies ES(x)510. b. If the number of subintervals used in the Midpoint Rule is ircreased by a factor of 3, the error is expected to decrease by a factor of 8. c. If the number of subintervals used in the Trapezoid Rule is increased by a factor of 4, the error is expected to decrease by a factcr of 16.Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given). 44. 0/2sin6xdx=532Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given). 45. 0/2cos9xdx=12831556E57E58E59EUsing Simpsons Rule Approximate the following integrals using Simpsons Rule. Experiment with values of n to ensure that the error is less than 103. 50. 0ln(2+cosx)dx=ln(2+32)61EPeriod of a pendulum A standard pendulum of length L swinging under only the influence of gravity (no resistance) has a period of T=40/2d1k2sin2, where 2 = g/L, k2 = sin2 (0/2), g 9.8 m/s2 is the acceleration due to gravity, and 0 is the initial angle from which the pendulum is released (in radians). Use numerical integration to approximate the period of a pendulum with L = 1 m that is released from an angle of 0 = /4 rad.Normal distribution of heights The heights of U.S. men are normally distributed with a mean of 69 inches and a standard deviation of 3 inches. This means that the fraction of men with a height between a and b (with a b) inches is given by the integral 132abe((x69)/3)2/2dx. What percentage of American men are between 66 and 72 inches tall? Use the method of your choice and experiment with the number of subintervals until you obtain successive approximations that differ by less than 103.64EU.S. oil produced and imported The figure shows the rate at which U.S. oil was produced and imported between 1920 and 2005 in units of millions of barrels per day. The total amount of oil produced or imported is given by the area of the region under the corresponding curve. Be careful with units because both days and years are used in this data set. a. Use numerical integration to estimate the amount of U.S. oil produced between 1940 and 2000. Use the method of your choice and experiment with values of n. b. Use numerical integration to estimate the amount of oil imported between 1940 and 2000. Use the method of your choice and experiment with values of n. (Source: U.S. Energy Information Administration)66EEstimating error Refer to Theorem 8.1 in the following exercises, Let f(x) = x3+1. a. Find a Midpoint Rule approximation to 16x3+1dx using n = 50 subintervals. b. Calculate f (x). c. Use the fact that f is decreasing and positive on [1, 6] to show that |f (x)| 15/(82) on [1, 6]. d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).Estimating error Refer to Theorem 7.2 and let f(x)=ex2. a. Find a Trapezoid Rule approximation to 01ex2dx using n = 50 subintervals. b. Calculate f(x). c. Explain why |f(x)| 18 on [0, 1], given that e 3. d. Use Theorem 7.2 to find an upper bound on the absolute error in the estimate found in part (a).Estimating error Refer to Theorem 7.2 and let f(x) = sin ex. a. Find a Trapezoid Rule approximation to 01sinexdx using n = 40 subintervals. b. Calculate f(x). c. Explain why |f(x)| 6 on [0, l], given that e 3 (Hint: Graph f.) d. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 7.2.Let f (x) = ex2 a. Find a Simpsons Rule approximation to 03ex2 dx using n = 30 subintervals. b. Calculate f (4)(x). c. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1 (Hint: Use a graph to find an upper bound for |f (4)(x)| on [0, 3].)71EExact Trapezoid Rule Prove that the Trapezoid Rule is exact (no error) when approximating the definite integral of a linear function.Arc length of an ellipse The length of an ellipse with axes of length 2a and 2b is 02a2cos2t+b2sin2tdt. Use numerical integration, and experiment with different values of n to approximate the length of an ellipse with a = 4 and b = 8.Sine integral The theory of diffraction produces the sine integral function Si(x)=0xsinttdt. Use the Midpoint Rule to approximate Si(1) and Si(10). (Recall that limx0sinxx=1.) Experiment with the number of subintervals until you obtain approximations that have an error less than 103. A rule of thumb is that if two successive approximations differ by less than 103, then the error is usually less than 103.Exact Simpsons Rule a. Use Simpsons Rule to approximate 04x3dx using two subintervals ( n = 2); compare the approximation to the value of the integral. b. Use Simpsons Rule to approximate 04x3dx using four subintervals ( n = 4); compare the approximation to the value of the integral. c. Use the error bound associated with Simpson s Rule given in Theorem 8.1 to explain why the approximations in parts (a) and (b) give the exact value of the integral. d. Use Theorem 8.1 to explain why a Simpsons Rule approximation using any (even) number of subintervals gives the exact value of abf(x)dx, where f(x) is a polynomial of degree 3 or less.Shortcut for the Trapezoid Rule Given a Midpoint Rule approximation M(n) and a Trapezoid Rule approximation T(n) for a continuous function on [a, b] with n subintervals, show that T(2n) = (T(n) + M(n))/2.Trapezoid Rule and concavity Suppose f is positive and its first two derivatives are continuous on [a, b] If f is positive on [a, b], then is a Trapezoid Rule estimate of abf(x) dx an underestimate or overestimate of the integral? Justify your answer using Theorem 8.1 and an illustration.Shortcut for Simpsons Rule Using the notation of the text, prove that S(2n)=4T(2n)T(n)3, for n 1.Another Simpsons Rule formula Another Simpsons Rule formula is S(2n)=2M(n)+T(n)3, for n 1. Use this rule to estimate 1e1/xdx using n = 10 subintervals.The function f(x) = 1 + x 1 decreases to 1 as x . Does 1f(x)dx exist?