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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Comparing a sine and a quadratic function Consider the functions f(x) = sin x and g(x)=42x(x). a. Carefully graph f and g on the same set of axes. Verify that both functions have a single local maximum on the interval [0, ] and that they have the same maximum value on [0, ]. b. On the interval [0, ], which is true: f(x) g(x), g(x) f(x),or neither? c. Compute and compare the average values of f and g on [0, ].Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume that f and g are even functions and p and q are odd functions. 53. aaf(g(x))dxSymmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume that f and g are even functions and p and q are odd functions. 54. aaf(p(x))dx51ESymmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume that f and g are even functions and p and q are odd functions. 56. aap(q(x))dx53EAlternative definitions of means Consider the function f(t)=abxt+1dxabxtdx. Show that the following means can be defined in terms of f. a. Arithmetic mean: f(0)=a+b2 b. Geometric mean: f(32)=ab c. Harmonic mean: f(3)=2aba+b d. Logarithmic mean: f(1)=balnblna (Source: Mathematics Magazine 78, 5, Dec 2005)Problems of antiquity Several calculus problems were solved by Greek mathematicians long before the discovery of calculus. The following problems were solved by Archimedes using methods that predated calculus by 2000 years. a. Show that the area of a segment of a parabola is 43 that of its inscribed triangle of greatest area. In other words, the area bounded by the parabola y = a2 x2 and the x-axis is 43 the area of the triangle with vertices (a, 0) and (0, a2). Assume that a 0 but is unspecified. b. Show that the area bounded by the parabola y = a2 x2 and the x-axis is 23 the area of the rectangle with vertices (a, 0) and ( a, a2). Assume that a 0 but is unspecified.56ESymmetry of powers Fill in the following table with either even or odd, and prove each result. Assume n is a nonnegative integer and fn means the nth power of f. f is even f is odd n is even fn is _____ fn is _____ n is odd fn is _____ fn is _____Bounds on an integral Suppose f is continuous on [a, b] with f(x) 0 on the interval. It can be shown that (ba)f(a+b2)abf(x)dx(ba)f(a)+f(b)2. a. Assuming f is nonnegative on [a, b], draw a figure to illustrate the geometric meaning of these inequalities. Discuss your conclusions. b. Divide these inequalities by (b a) and interpret the resulting inequalities in terms of the average value of f on [a, b].Generalizing the Mean Value Theorem for Integrals Suppose f and g are continuous on [a, b] and let h(x)=(xb)axf(t)dt+(xa)xbg(t)dt. a. Use Rolles Theorem to show that there is a number c in (a, b) such that acf(t)dt+cbg(t)dt=f(c)(bc)+g(c)(ca), which is a generalization of the Mean Value Theorem for Integrals. b. Show that there is a number c in (a, b) such that acf(t)dt=f(c)(bc). c. Use a sketch to interpret part (b) geometrically. d. Use the result of part (a) to give an alternative proof of the Mean Value Theorem for Integrals. (Source: The College Mathematics Journal, 33, 5, Nov 2002)A sine integral by Riemann sums Consider the integral I=0/2sinxdx. a. Write the left Riemann sum for I with n subintervals. b. Show that lim0(cos+sin12(1cos))=1. c. It is a fact that k0n1sin(k2n)=cos(2n)+sin(2n)12(1cos(2n)). Use this fact and part (b) to evaluate I by taking the limit of the Riemann sum as n .Find a new variable u so that 4x3(x4+5)10dx=u10du.In Example 2a, explain why the same substitution would not work as well for the integral x3(x5+6)9dx. Example 2 Introducing a Constant Find the following indefinite integrals. a.x4(x5+6)9dx b.cos3xsinxdxEvaluate cos6xdxwithout using the substitution method.Evaluate 44x2dx.Changes of variables occur frequently in mathematics. For example, suppose you want to solve the equation x4 13x2 + 36 = 0. If you use the substitution u = x2, what is the new equation that must be solved for u? What are the roots of the original equation?Review Questions 1. On which derivative rule is the Substitution Rule based?Why is the Substitution Rule referred to as a change of variables?The composite function f(g(x)) consists of an inner function g and an outer function f. If an integrand includes f(g(x)), which function is often a likely choice for a new variable u?Find a suitable substitution for evaluating tanxsec2xdx and explain your choice.When using a change of variables u = g(x) to evaluate the definite integral abf(g(x))g(x)dx, how are the limits of integration transformed?If the change of variables u = x2 4 is used to evaluate the definite integral 24f(x)dx, what are the new limits of integration?Substitution given Use the given substitution to find the following indefinite integrals. Check your answer by differentiating. 13. 2x(x2+1)4dx,u=x2+1Substitution given Use the given substitution to find the following indefinite integrals. Check your answer by differentiating. 14. 8xcos(4x2+3)dx,u=4x2+3Substitution given Use the given substitution to find the following indefinite integrals. Check your answer by differentiating. 15. sin3xcosxdx,u=sinxSubstitution given Use the given substitution to find the following indefinite integrals. Check your answer by differentiating. 16. (6x+1)3x2+xdx,u=3x2+xUse a substitution of the form u = ax + b to evaluate the following indefinite integrals. 11.(x+1)12dxUse a substitution of the form u = ax + b to evaluate the following indefinite integrals. 12.e3x+1dxUse a substitution of the form u = ax + b to evaluate the following indefinite integrals. 13.2x+1dxUse a substitution of the form u = ax + b to evaluate the following indefinite integrals. 14.cos(2x+5)dxUse Table 5.6 to evaluate the following indefinite integrals. a.e10xdx b.sec5xtan5xdx c.sin7xdx d.cosx7dx e.dx81+9x2 (Hint: Factor a 9 out of the denominator first.) f.dx36x2Use Table 5.6 to evaluate the following definite integrals. a.0110xdx b.0/40cos20xdx c.326dxxx29 d.0/16sec24xdxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 17.2x(x21)99dxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 18.xex2dxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 19.2x214x3dxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 20.(x+1)42xdxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 21.(x2+x)10(2x+1)dxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 22.110x3dxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 23.x3(x4+16)6dxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 24.sin10cosdIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 25.dx364x2Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 26.dx19x2Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 27.6x24x3dxx9sinx10dxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 29.(x63x2)4(x5x)dxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 30.dx1+4x2Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 31.3125x2dxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 32.2x4x21dx,x12Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 33.ew36+e2wdwIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 34.8x+62x2+3xdxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 35.xcscx2cotx2dxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 36.sec4wtan4wdwsec2(10x+7)dxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 38.tan1ww2+1dwIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 39.104t+1dtIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 40.(sin5x+3sin3xsinx)cosxdxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 41.csc2xcot3xdxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 42.(x3/2+8)5xdxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 43.sinxsec8xdxIndefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. 44.e2xe2x+1dxDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 45.0/8cos2xdxDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 46.012e2xdxDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 47. 012x(4x2)dxDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 48. 022x(x2+1)2dxDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 49. 132x2x+4dxDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 50. 22cos8dDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 51. 0/2sin2cosdDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 52. 0/4sinxcos2xdxDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 53. ln4ln2ewcosewdwDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals 54. /16/88csc24xdxDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals 55. 12x2ex3+1dxDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals 56. 04p9+p2dpDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 57. /4/2cosxsin2xdxDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 58. 0/4sincos3dDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 59. 2/(53)2/5dxx25x21Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 60. 01v3+1v4+4v+4dvDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 61. 04xx2+1dxDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 62. 01/8x116x2dxDefinite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. 63. 1/31/349x2+1dx0ln4ex3+2exdx01x1x2dx66E67E06/5dx25x2+3602x316x4dx11(x1)(x22x)7dx0sinx2+cosxdx01(v+1)(v+2)2v3+9v2+12v+36dv1249x2+6x+1dx0/4esin2xsin2xdxAverage velocity An object moves in one dimension with a velocity in m/s given by v(t)=8sint+2t. Find its average velocity over the time interval from t=0 to t=10. where t measured in seconds.Periodic motion An object moves along a line with a velocity in m/s given by v(t) = 8 cos (t/6). Its initial position is s(0) = 0. a. Graph the velocity function. b. As discussed in Chapter 6, the position of the object is given by s(t)=0tv(y)dy, for t 0. Find the position function, for t 0. c. What is the period of the motionthat is, starting at any point, how long does it take the object to return to that position?Population models The population of a culture of bacteria has a growth rate given by p(t)=200(t+1)r bacteria per hour, for t 0, where r 1 is a real number. In Chapter 6 it is shown that the increase in the population over the time interval [0, t] is given by 0tp(s)ds. (Note that the growth rate decreases in time, reflecting competition for space and food.) a. Using the population model with r = 2, what is the increase in the population over the time interval 0 t 4? b. Using the population model with r = 3, what is the increase in the population over the time interval 0 t 6? c. Let P be the increase in the population over a fixed time interval [0, T]. For fixed T, does P increase or decrease with the parameter r? Explain. d. A lab technician measures an increase in the population of 350 bacteria over the 10-hr period [0, 10]. Estimate the value of r that best fits this data point. e. Looking ahead: Use the population model in part (b) to find the increase in population over the time interval [0, T], for any T 0. If the culture is allowed to grow indefinitely (T ), does the bacteria population increase without bound? Or does it approach a finite limit?Variations on the substitution method Evaluate the following integrals. 78. xx2dxVariations on the substitution method Find the following integrals. 33. xx4dxVariations on the substitution method Find the following integrals. 34. y2(y+1)4dyVariations on the substitution method Find the following integrals. 35. xx+43dxVariations on the substitution method Find the following integrals. 36. exexex+exdxVariations on the substitution method Find the following integrals. 37. x2x+13dxVariations on the substitution method Find the following integrals. 38. (z+1)3z+2dzx(x+10)9dx033dx9+x2Integrals with sin2 x and cos2 x Evaluate the following integrals. 53. cos2xdxIntegrals with sin2 x and cos2 x Evaluate the following integrals. 54. sin2xdxIntegrals with sin2 x and cos2 x Evaluate the following integrals. 55. sin2(+6)dIntegrals with sin2 x and cos2 x Evaluate the following integrals. 56. 0/4cos28dIntegrals with sin2 x and cos2 x Evaluate the following integrals. 57. /4/4sin22dIntegrals with sin2 x and cos2 x Evaluate the following integrals. 58. xcos2(x2)dxIntegrals with sin2 x and cos2 x Evaluate the following integrals. 59. 0/6sin2ysin2y+2dy(Hint: sin 2y = 2 sin y cos y.)94EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume that f, f, and f are continuous functions for all real numbers. a. f(x)f(x)dx=12(f(x))2+C. b. (f(x))nf(x)dx=1n+1(f(x))n+1+C,n1. c. sin2xdx=2sinxdx. d. (x2+1)9dx=(x2+1)1010+C. e. abf(x)f(x)dx=f(b)f(a).96E97EAreas of regions Find the area of the following regions. 81. The region bounded by the graph of f(x) = (x 4)4 and the x-axis between x = 2 and x = 699E100ESubstitutions Suppose that p is a nonzero real number and f is an odd integrable function with 01f(x)dx=. Evaluate each integral. a. 0/(2p)(cospx)f(sinpx)dx b. /2/2(cosx)f(sinx)dx102EAverage value of sine functions Use a graphing utility to verify that the functions f(x) = sin kx have a period of 2/k, where k = 1, 2, 3, . Equivalently, the first hump of f(x) = sin kx occurs on the interval [0, /k]. Verify that the average value of the first hump of f(x) = sin kx is independent of k. What is the average value?Equal areas The area of the shaded region under the curve y = 2 sin 2x in (a) equals the area of the shaded region under the curve y = sin x in (b). Explain why this is true without computing areas.Equal areas The area of the shaded region under the curve y=(x1)22x on the interval [4, 9] in (a) equals the area of the shaded region under the curve y = x2 on the interval [1, 2] in (b). Without computing areas, explain why.106E107E108EMore than one way Occasionally, two different substitutions do the job. Use each substitution to evaluate the following integrals. 99. 01xx+adx;a0(u=x+aandu=x+a)110E111Esin2 ax and cos2 ax integrals Use the Substitution Rule to prove that sin2axdx=x2sin(2ax)4a+Candcos2axdx=x2+sin(2ax)4a+C.Integral of sin2 x cos2 x Consider the integral I=sin2xcos2xdx. a. Find I using the identity sin 2x = 2 sin x cos x. b. Find I using the identity cos2 x = 1 sin2 x. c. Confirm that the results in parts (a) and (b) are consistent and compare the work involved in each method.Substitution: shift Perhaps the simplest change of variables is the shift or translation given by u = x + c, where c is a real number. a. Prove that shifting a function does not change the net area under the curve, in the sense that abf(x+c)dx=a+cb+cf(u)du. b. Draw a picture to illustrate this change of variables in the case that f(x) = sin x, a = 0, b = , and c = /2.115E116E117E118EMultiple substitutions If necessary, use two or more substitutions to find the following integrals. 111. 0/2cossincos2+16d (Hint: Begin with u = cos .)Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume f and f are continuous functions for all real numbers. a. If A(x)=axf(t)dt, and f(t) = 2t 3, then A is a quadratic function. b. Given an area function A(x)=axf(t)dt and an antiderivative F of f, it follows that A(x) = F(x). c. abf(x)dx=f(b)f(a). d. If f is continuous on [a, b] and abf(x)dx=0, then f(x) = 0 on [a, b]. e. If the average value of f on [a, b] is zero, then f(x) = 0 on [a, b]. f. ab(2f(x)3g(x))dx=2abf(x)dx+3bag(x)dx. g. f(g(x))g(x)dx=f(g(x))+C.2REAscent rate of a scuba diver Divers who ascend too quickly in the water risk decompression illness. A common recommendation for a maximum rate of ascent is 30 feet/minute with a 5-minute safety stop 15 feet below the surface of the water. Suppose that a diver ascends to the surface in 8 minutes according to the velocity function v(t)={30if0t20if2t715if7t8. a. Graph the velocity function v. b. Compute the area under the velocity curve. c. Interpret the physical meaning of the area under the velocity curve.Use the tabulated values of f to estimate the value of 06f(x)dxby evaluating the left, right, and midpoint Riemann sums using a regular partition with n=3 subintervals.Estimate 144x+1dx by evaluating the left, right, and midpoint Riemann sums using a regular partition with n=6 subintervals6REEstimating a definite integral Use a calculator and midpoint Riemann sums to approximate 1252x1dx. Present your calculations in a table showing the approximations for n=10, 30. and 60 subintervals, assuming a regular partition. Make a conjecture about the exact value of the integral and verify your conjecture using the Fundamental Theorem of Calculus.Suppose the expression lim0k=1n(xk3+xk)xk is the limit of a Riemann sum of a function f on [3, 8]. Identify a possible function f and express the limit as a definite integral.Integration by Riemann sums Consider the integral 14(3x2)dx. a. Evaluate the right Riemann sum for the integral with n = 3. b. Use summation notation to express the right Riemann sum in terms of a positive integer n. c. Evaluate the definite integral by taking the limit as n of the Riemann sum of part (b). d. Confirm the result of part (c) by graphing y = 3x 2 and using geometry to evaluate the integral. Then evaluate 14(3x2)dx with the Fundamental Theorem of Calculus.Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. 8. 01(4x2)dxLimit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. 9. 02(x24)dxLimit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. 10. 12(3x2+x)dx13RESum to integral Evaluate the following limit by identifying the integral that it represents: limnk=1n((4kn)5+1)(4n).Symmetry properties Suppose that 04f(x)dx=10 and 04g(x)dx=20. Furthermore, suppose that f is an even function and g is an odd function. Evaluate the following integrals. a. 44f(x)dx b. 443g(x)dx c. 44(4f(x)3g(x))dx d. 018xf(4x2)dx e. 223xf(x)dxProperties of integrals The figure shows the areas of regions bounded by the graph of f and the x-axis. Evaluate the following integrals.
Properties of integrals Suppose that 14f(x)dx=6, and14g(x)dx=4, and 34f(x)dx=2. Evaluate the following integrals or state that there is not enough information. 39. 143f(x)dxProperties of integrals Suppose that 14f(x)dx=6, and14g(x)dx=4, and 34f(x)dx=2. Evaluate the following integrals or state that there is not enough information. 40. 412f(x)dxProperties of integrals Suppose that 14f(x)dx=6, and14g(x)dx=4, and 34f(x)dx=2. Evaluate the following integrals or state that there is not enough information. 41. 14(3f(x)2g(x))dxProperties of integrals Suppose that 14f(x)dx=6, and14g(x)dx=4, and 34f(x)dx=2. Evaluate the following integrals or state that there is not enough information. 42. 14f(x)g(x)dxProperties of integrals Suppose that 14f(x)dx=6, and14g(x)dx=4, and 34f(x)dx=2. Evaluate the following integrals or state that there is not enough information. 43. 13f(x)g(x)dxProperties of integrals Suppose that 14f(x)dx=6, and14g(x)dx=4, and 34f(x)dx=2. Evaluate the following integrals or state that there is not enough information. 44. 41(f(x)=g(x))dxArea by geometry Use geometry to evaluate the following definite integrals, where the graph of f is given in the figure. a. 04f(x)dx b. 64f(x)dx c. 57f(x)dx d. 07f(x)dxDisplacement by geometry Use geometry to find the displacement of an object moving along a line for the time intervals (i) 0 t 5, (ii) 3 t 7, and (iii) 0 t 8, where the graph of its velocity v = g(t) is given in the figure.Area by geometry Use geometry to evaluate 048xx2dx. (Hint: Complete the square.)Use geometry and properties of integrals to evaluate the following definite integrals. 26. 40(2x+16x2)dx (Hint: Write the integral as a sum of two integrals.)27RE28REEvaluate the following derivatives. 29. ddx7x1+t4+t6dtEvaluate the following derivatives. 30.ddx3excost2dtEvaluate the following derivatives. 31. ddxx5sinw6dwEvaluate the following derivatives. 32. ddxx25sinw6dwEvaluate the following derivatives. 33. ddxxxdtt10+1Evaluate the following derivatives. 34. ddxx2exsin3tdtFind the intervals on which f(x)=x1(t3)(t6)11dt is increasing and the intervals on which it is decreasingArea function by geometry Use geometry to find the area A(x) that is bounded by the graph of f(t) = 2t 4 and the t-axis between the point (2, 0) and the variable point (x, 0), where x 2. Verify that A(x) = f(x).Given that F=f, use the substitution method to show that f(ax+b)dx=1aF(ax+b)+C, for nonzero constants a and b.Evaluating integrals Evaluate the following integrals. 38. 15dxEvaluating integrals Evaluate the following integrals. 15. 22(3x42x+1)dxEvaluating integrals Evaluate the following integrals. 18. 01(4x212x16+1)dxEvaluating integrals Evaluate the following integrals. 19. (9x87x6)dxEvaluating integrals Evaluate the following integrals. 42. x+1xdxEvaluating integrals Evaluate the following integrals. 21. 01x(x+1)dxEvaluating integrals Evaluate the following integrals. 44. (3x+1)(3x2+2x+1)3dxEvaluating integrals Evaluate the following integrals. 45. /6/3(sec2t+csc2t)dtEvaluating integrals Evaluate the following integrals. 46. /12/9(cos3xcot3x+sec3xtan3x)dxEvaluating integrals Evaluate the following integrals. 47. 22dxxx21Evaluating integrals Evaluate the following integrals. 48. 14(v+vv)dvEvaluating integrals Evaluate the following integrals. 49. cosxsin7/4xdxEvaluating integrals Evaluate the following integrals. 50. 1edxx(1+lnx)Evaluating integrals Evaluate the following integrals. 51. x2cosx3dxEvaluating integrals Evaluate the following integrals. 52. cos3t1+sin3tdtEvaluating integrals Evaluate the following integrals. 53. cos7w16+sin27wdwEvaluating integrals Evaluate the following integrals. 54. 1+tan2tsec22tdtEvaluating integrals Evaluate the following integrals. 55. 01x2x2+1dxEvaluating integrals Evaluate the following integrals. 16. cos3xdxEvaluating integrals Evaluate the following integrals. 57. 02(2x+1)3dxEvaluating integrals Evaluate the following integrals. 20. 22e4x+8dx015re3r2+2drsinzsin(cosz)dzex+exdxEvaluating integrals Evaluate the following integrals. 22. y2y3+27dydx14x2Evaluating integrals Evaluate the following integrals. 24. y2(3y3+1)4dy02cos2x6dxEvaluating integrals Evaluate the following integrals. 26. xsinx2cos8x2dxEvaluating integrals Evaluate the following integrals. 27. 0sin25dEvaluating integrals Evaluate the following integrals. 28. 0(1cos23)dEvaluating integrals Evaluate the following integrals. 29. 23x2+2x2x3+3x26xdxEvaluating integrals Evaluate the following integrals. 30. 0ln2ex1+e2xdx71RE33(511x17+302x13+117x9+303x3+x2)dx1x2sin1xdx(tan1x)51+x2dxdx(tan1x)(1+x2)sin1x1x2dxx(x+3)10dxx7x4+1dxEvaluating integrals Evaluate the following integrals. 25. 03x25x2dxEvaluating integrals Evaluate the following integrals. 23.01dx4x22/52/5dxx25x21sin2x1+cos2xdx (Hint: sin2x=2sinxcosx.)1010x200x2dx/2/2(cos2x+cosxsinx3sinx5)dx04f(x)dx for f(x)={2x+1ifx33x2+2x8ifx305|2x8|dx87REArea of regions Compute the area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region. 32. f(x) = x3 x on [1, 0]89RE90RE91REArea versus net area Find (i) the net area and (ii) the area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region. 36. f(x) = x2 x on [0, 3]Gateway Arch The Gateway Arch in St Louis is 630 ft high and has a 630-ft base Its shape can be modeled by the function y=1260315(e0.00418x+e0.00418x), where the base of the arch is [315, 315] and x and y are measured in feet. Find the average height of the arch above the ground.Root mean square The root mean square (or RMS) is another measure of average value, often used with oscillating functions (for example, sine and cosine functions that describe the current, voltage, or power in an alternating circuit). The RMS of a function f on the interval [0. T] is fRMS=1T0Tf(t)2dt. Compute the RMS of f(t)=Asin(t). where A and are positive constants and T is any integer multiple of the period of f. which is 2/.Displacement from velocity A particle moves along a line with a velocity given by v(t) = 5 sin t starting with an initial position s(0) = 0. Find the displacement of the particle between t = 0 and t = 2, which is given by s(t)=02v(t)dt. Find the distance traveled by the particle during this interval, which is 02v(t)dt.Velocity to displacement An object travels on the x-axis with a velocity given by v(t) = 2t + 5, for 0 t 4. a. How far does the object travel, for 0 t 4? b. What is the average value of v on the interval [0, 4]? c. True or false: The object would travel as far as in part (a) if it traveled at its average velocity (a constant), for 0 t 4.Find the average value of f(x)=e2xon [0, ln 2].Average height A baseball is launched into the outfield on a parabolic trajectory given by y = 0.01x(200 x). Find the average height of the baseball over the horizontal extent of its flight.Average values Integration is not needed. a. Find the average value of f shown in the figure on the interval [1, 6] and then find the point(s) c in (1, 6) guaranteed to exist by the Mean Value Theorem for Integrals. b. Find the average value of f shown in the figure on the interval [2, 6] and then find the point(s) c in (2, 6) guaranteed to exist by the Mean Value Theorem for Integrals.100REAn unknown function Assume f is continuous on [2, 4], 12f(2x)dx=10,and f(2) = 4. Evaluate f(4).102RE103REChange of variables Use the change of variables u3 = x2 1 to evaluate the integral 13xx213dx.105REArea functions and the Fundamental Theorem Consider the function f(t)={tif2t0t22if0t2 and its graph shown below. Let F(x)=1xf(t)dt and 2xf(t)dt. 56. a. Evaluate G(1) and G(1). b. Use the Fundamental Theorem to find an expression for G(x), for 2 x 0. c. Use the Fundamental Theorem to find an expression for G(x), for 0 x 2. d. Evaluate G(0) and G(1). Interpret these values. e. Find a constant C such that F(x) = G(x) + C.Limits with integrals Evaluate the following limits. 57. limx22xet2dtx2Limits with integrals Evaluate the following limits. 58. limx11x2et3dtx1109REArea with a parameter Let a 0 be a real number and consider the family of functions f(x) = sin ax on the interval [0, /a]. a. Graph f, for a = 1, 2, 3. b. Let g(a) be the area of the region bounded by the graph of f and the x-axis on the interval [0, /a]. Graph g for 0 a . Is g an increasing function, a decreasing function, or neither?Inverse tangent integral Prove that for nonzero constants a and b, dxa2x2+b2=1abtan1(axb)+C112RE113REExponential inequalities Sketch a graph of f(t) = et on an arbitrary interval [a, b]. Use the graph and compare areas of regions to prove that e(a+b)/2ebeabaea+eb2.Equivalent equations Explain why if a function u satisfies the equation u(x)+20xu(t)dt=10, then it also satisfies the equation u(x) + 2u(x) = 0. Is it true that if u satisfies the second equation, then it satisfies the first equation?Unit area sine curve Find the value of c such that the region bounded by y=csinxand the x-axis on the interval [0, ] has area 1.Unit area cubic Find the value of c0 such that the region bounded by the cubic y=x(xc)2 and the x-axis on the interval [0. c] has area 1.A police officer leaves his station on a north-south freeway at 9 a.m., travelling north (the positive direction) for 40 mi between 9 a.m. and 10 a.m. From 10 a.m. to 11 a.m., he travels south to a point 20 mi south of the station. What are the distance traveled and the displacement between 9 a.m. and 11 a m.?Describe a possible motion of an object along a line, for 0t5, for which the displacement and the distance traveled are different.Is the position s(t) a number or a function? For fixed times t = a and t = b, is the displacement s(b)s(a) a number or a function?Without doing further calculations, what are the displacement and distance traveled by the block in Example 2 over the interval [0, 2]? Example 2 Position from Velocity A block hangs at rest from a massless spring at the origin (s=0). At t=0, the block is pulled downward 14m to its initial position s(0)=14 and released (Figure 6.4). Its velocity (in m/s) is given by v(t)=14sint for t0. Assume the upward direction is positive. a. Find the position of the block, for t0. b. Graph the position function, for 0t3. c. When does the block move through the origin 'or the first time? d. When does the block reach its highest point for the first time and what is its position at that time? When does the block return to its lowest point?Suppose (unrealistically) in Example 3 that the velocity of the skydiver is 80 m/s. for , and then it changes instantaneously to 6 m/s. for . Sketch the velocity function and, without integrating, find the distance the skydiver falls in 40 s. Example 3 Skydiving Suppose a skydiver leaps from a hovering helicopter and falls in a straight line. Assume he reaches a terminal velocity of 80 m/s immediately at t = 0 and falls for 19 seconds, at which time he opens his parachute. The velocity decreases linearly to 6 m/s over a two-second period and then remains constant until he reaches the ground at t = 40 s. The motion is described by the velocity function v(t)={80if0t1978337tif19t216if21t40 Determine the height above the ground from which the skydiver jumpedIs the cost of increasing production from 0000 books to 12 000 books in Example 6 more or less than the cost of increasing production from 12.000 books to 15.000 books? Explain. Example 6 Production Costs A book publisher estimates that the marginal cost of producing a particular title (in dollars/book) is given by C(x)=120.0002x, where 0x50,000 is the number of books printed. What is the cost of producing the 12,001st through the 15,000th book?Explain the meaning of position, displacement, and distance traveled as they apply to an object moving along a line.Suppose the velocity of an object moving along a line is positive. Are displacement and distance traveled equal? Explain.Given the velocity function v of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.Explain how to use definite integrals to find the net change in a quantity, given the rate of change of that quantity.5EWhat is the result of integrating a population growth rate between times t = a and t = b, where b a?Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a Line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given. a. On what intervals is the object moving in the positive direction? b. What is the displacement of the object over the interval [0, 3]? c. What is the total distance traveled by the object over the interval [1, 5]? d. What is the displacement of the object over the interval [0, 5]? e. Describe the position of the object relative to its initial position after 5 hours.Displacement and distance from velocity Consider the velocity function shown below of an object moving along a line. Assume time is measured in seconds and distance is measured in meters. The areas of four regions bounded by the velocity curve and the f-axis are also given. a. On what intervals is the object moving in the negative direction? b. What is the displacement of the object over the interval [2, 6]? c. What is the total distance traveled by the object over the interval [0, 6]? d. What is the displacement of the object over the interval [0, 8]? e. Describe the position of the object relative to its initial position after 8 seconds.Velocity graphs The figures show velocity functions for motion along a straight line. Assume the motion begins with an initial position of s(0) = 0. Determine the following: a. The displacement between t = 0 and t = 5 b. The distance traveled between t = 0 and t = 5 c. The position at t = 5 d. A piecewise function for s(t) 51.Velocity graphs The figures show velocity functions for motion along a straight line. Assume the motion begins with an initial position of s(0) = 0. Determine the following: a. The displacement between t = 0 and t = 5 b. The distance traveled between t = 0 and t = 5 c. The position at t = 5 d. A piecewise function for s(t) 50.Distance traveled and displacement Suppose an object moves along a line with velocity (in m/s) v(t)=3sin2t, for 0t2, where t is measured in seconds (see figure). a. Find the distance traveled by the object on the time interval [0, /2]. b. Find the displacement of the object on the time intervals (0, /2], [0, ], (0, 3 /2], and [0, 2]. (Hint: Use your answer to part (a) together with the symmetry of the graph to find the displacement values.) c. Find the distance traveled by the object on the time interval [0, 2].Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=62t, for 0t6, where t is measured in seconds. a. Graph the velocity function on the interval 0t6. Determine when the motion is in the positive direction and when it is in the negative direction on 0t6. b. Find the displacement of the object on the interval 0t6. c. Find the distance traveled by the object on the interval 0t6.Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s. a. Determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. 13. v(t)=3t26t on [0, 3]Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s. a. Determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. 14. v(t)=4t324t2+20t on [0, 5]Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s. a. Determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. 15. v(t)=3t218t+24 on [0, 5]Displacement from velocity Assume t is time measured in seconds and velocities have units of m/s. a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. 14. v(t) = 50e2t on 0 t 4Position from velocity Consider an object moving along a line with the given velocity v and initial position. a. Determine the position function, for t0, using the antiderivative method b. Determine the position function, for t0, using the Fundamental Theorem of Calculus (Theorem 6.1). Check for agreement with the answer to part (a). 17. v(t)=sint on [0, 2]; s(0)=1Position from velocity Consider an object moving along a line with the given velocity v and initial position. a. Determine the position function, for t0, using the antiderivative method b. Determine the position function, for t0, using the Fundamental Theorem of Calculus (Theorem 6.1). Check for agreement with the answer to part (a). 18. v(t)=t3+3t22t on [0, 3]; s(0)=4Position from velocity Consider an object moving along a line with the given velocity v and initial position a. Determine the position function, for t0, using the antiderivative method b. Determine the position function, for t0, using the Fundamental Theorem of Calculus (Theorem 6.1). Check for agreement with the answer to part (a). 19. v(t)=62t on [0, 5]; s(0)=0Position from velocity Consider an object moving along a line with the given velocity v and initial position a. Determine the position function, for t 0, using the antiderivative method b. Determine the position function, for t 0, using the Fundamental Theorem of Calculus (Theorem 6.1). Check for agreement with the answer to part (a). 20.v(t)=3sinton[0,4];s(0)=1Position from velocity Consider an object moving along a line with the given velocity v and initial position a. Determine the position function, for t 0, using the antiderivative method b. Determine the position function, for t 0, using the Fundamental Theorem of Calculus (Theorem 6.1). Check for agreement with the answer to part (a). 21.v(t)=9t2on[0,4];s(0)=2Position from velocity Consider an object moving along a line with the given velocity v and initial position a. Determine the position function, for t 0, using the antiderivative method b. Determine the position function, for t 0, using the Fundamental Theorem of Calculus (Theorem 6.1). Check for agreement with the answer to part (a). 22.v(t)=1t+1on[0,8];s(0)=4Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2 cos t, for t 0. Assume that the positive direction is upward and that s(0) = 0. a. Determine the position function, for t 0. b. Graph the position function on the interval [0, 4]. c. At what times does the mass reach its low point the first three times? d. At what times does the mass reach its high point the first three times?Cycling distance A cyclist rides down a long straight road at a velocity (in m/min) given by v(t) = 400 20t, for 0 t 10, where t is measured in minutes. a. How far does the cyclist travel in the first 5 min? b. How far does the cyclist travel in the first 10 min? c. How far has the cyclist traveled when her velocity is 250 m/min?Flying into a headwind The velocity (in mi/hr) of an airplane flying into a headwind is given by v(t) = 30(16 t2), for 0 t 3. Assume that s(0) = 0 and t is measured in hours. a. Determine and graph the position function, for 0 t 3. b. How far does the airplane travel in the first 2 hr? c. How far has the airplane traveled at the instant its velocity reaches 400 mi/hr?Day hike The velocity (in mi/hr) of a hiker walking along a straight trail is given by v(t)=3sin2t2, for 0t4. Assume s(0) = 0 and t is measured in hours a.Determine the position function, for 0t4. (Hint: sin2t=1cos2t2.) b.What is the distance traveled by the hiker in the first 15 min of the hike? c.What is the hikers position at t = 3?Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function v(t)={3tif0t2060if20t45240ift45, where t is measured in seconds and v has units of m/s. a. Graph the velocity function, for 0 t 70. When is the velocity a maximum? When is the velocity zero? b. What is the distance traveled by the automobile in the first 30 s? c. What is the distance traveled by the automobile in the first 60 s? d. What is the position of the automobile when t = 75?Probe speed A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by v(t) = 9.8t, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean. a. Graph the velocity function. b. How far does the probe fall in the first 30 s after it is released? c. If the probe was released from an altitude of 3 km, w hen does it enter the ocean?Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2). 29.a(t)=32;v(0)=70;s(0)=10Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2). 30.a(t)=32;v(0)=50;s(0)=0Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2). 31.a(t)=98;v(0)=20;s(0)=0Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2). 32.a(t)=et;v(0)=60;s(0)=40Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2). 33.a(t)=0.01t;v(0)=10;s(0)=0Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2). 34.a(t)=20(t+2)2;v(0)=20;s(0)=10Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2). 35.a(t)=cos2t;v(0)=5;s(0)=7Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2). 36.a(t)=2t(t2+1)2;v(0)=0;s(0)=0Acceleration A drag racer accelerates at a(t) = 88 ft/s2. Assume v(0) = 0, s(0) = 0, and t is measured in seconds. a. Determine the position function, for t 0. b. How far does the racer travel in the first 4 seconds? c. At this rate, how long will it take the racer to travel 14 mi? d. How long does it take the racer to travel 300 ft? e. How far has the racer traveled when it reaches a speed of 178 ft/s?Deceleration A car slows down with an acceleration of a(t) = 15 ft/s2 Assume v(0) = 60 ft/s, s(0) = 0: and t is measured in seconds a.Determine the position function, for t 0 b.How far does the car travel in the time it takes to come to rest?Approaching a station At t = 0, a train approaching a station begins decelerating from a speed of 80 mi/hr according to the acceleration function a(t) = 1280(1 + 8t)3, where t 0 is measured in hours. How far does the train travel between t = 0 and t = 0.2? Between t = 0.2 and t = 0.4? The units of acceleration are mi/hr2.Population growth 40. Starting with an initial value of P(0) = 55, the population of a prairie dog community grows at a rate of P(t) = 20 t/5 (prairie dogs/month), for 0 t 200, where t is measured in months. a. What is the population 6 months later? b. Find the population P(t), for 0 t 200.