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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Consider the linear function f(x) = 2x + 5 and the region bounded by its graph and the x-axis on the interval [3, 6]. Suppose the area of this region is approximated using midpoint Riemann sums. Then the approximations give the exact area of the region for any number of subintervals. b. A left Riemann sum always overestimates the area of a region bounded by a positive increasing function and the x-axis on an interval [a, b]. c. For an increasing or decreasing nonconstant function on an interval [a, b] and a given value of n, the value of the midpoint Riemann sum always lies between the values of the left and right Riemann sums.60E61E62E63E64EIdentifying Riemann sums Fill in the blanks with an interval and a value of n. 65. k=14f(1+k)1 is a right Riemann sum for f on the interval [_____, _____] with n = _____.Identifying Riemann sums Fill in the blanks with an interval and a value of n. 66. k=14f(2+k)1 is a right Riemann sum for f on the interval [_____, _____] with n = _____.67E68EApproximating areas Estimate the area of the region bounded by the graph of f(x) = x2 + 2 and the x-axis on [0, 2] in the following ways. a. Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically. b. Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a midpoint Riemann sum. Illustrate the solution geometrically. c. Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure). a.Describe the motion of the object over the interval [0, 6]. b.Use geometry to find the displacement of the object between t = 0 and t = 3. c.Use geometry to find the displacement of the object between t = 3 and t = 5. d.Assuming the velocity remains 30 m/s, for t 4, find the function that gives the displacement between t = 0 and any time t 4.Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure). a. Describe the motion of the object over the interval [0, 6]. b. Use geometry to find the displacement of the object between t = 0 and t = 2. c. Use geometry to find the displacement of the object between t = 0 and t = 2. d. Assuming that the velocity remains 10 m/s, for t 5, find the function that gives the displacement between t = 0 and any time t 5.Flow rates Suppose a gauge at the outflow of a reservoir measures the flow rate of water in units of ft3/hr. In Chapter 6. we show that the total amount of water that flows out of the reservoir is the area under the flow rate curve. Consider the flow-rate function shown in the figure. a. Find the amount of water (in units of ft3) that flows out of the reservoir over the interval [0, 4]. b. Find the amount of water that flows out of the reservoir over the interval [8, 10]. c. Does more water flow out of the reservoir over the interval [0, 4] or [4, 6]? d. Show that the units of your answer are consistent with the units of the variables on the axes.Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6. we show that the mass of the rod is the area under the density curve. a. Find the mass of the left half of the rod (0 x 5). b. Find the mass of the right half of the rod (5 x 10). c. Find the mass of the entire rod (0 x 10). d. Find the point along the rod at which it will balance (called the center of mass).74E75E76E77ERiemann sums for constant functions Let f(x) = c, where c 0, be a constant function on [a, b]. Prove that any Riemann sum for any value of n gives the exact area of the region between the graph of f and the x-axis on [a, b].79E80E81ESuppose f(x) = 5. What is the net area of the region bounded by the graph of f and the x-axis on the interval [1, 5]? Make a sketch of the function and the region.Sketch a continuous function f that is positive over the interval [0, 1) and negative over the interval (1, 2], such that the net area of the region bounded by the graph of f and the x-axis on [0, 2] is zero.Graph f(x) = x and use geometry to evaluate 11xdx.Let f(x) = 5 and use geometry to evaluate 13f(x)dx. What is the value of abcdx, where c is a real number?Evaluate abf(x)dx+baf(x)dx assuming f is integrate on [a, b].Evaluate 12xdx and 12|x|dx using geometry.What does net area measure?Under what conditions does the net area of a region (bounded by a continuous function) equal the area of a region? When does the net area of a region differ from the area of a region?3EUse the graph of y = g(x) to estimate 210g(x)dx using a left, right, and midpoint Riemann sum with n = 4.Suppose f is continuous on [2, 8]. Use the table of values of f to estimate 28f(x)dx using a left, right, and midpoint Riemann sum with n = 3.Suppose g is continuous on [1, 9]. Use the table of values of g to estimate 19g(x)dx using a left, right, and midpoint Riemann sum with n = 4.Sketch a graph of y = 2 on [1, 4] and use geometry to find the exact value of 142dx.Sketch a graph of y = 3 on [1, 5] and use geometry to find the exact value of 15(3)dx.Sketch a graph of y = 2x on [1, 2] and use geometry to find the exact value of 122xdx.Suppose 13f(x)dx=10 and 13g(x)dx=20. Evaluate 13(2f(x)4g(x))dx and 31(2f(x)4g(x))dx.Use graphs to evaluate 02sinxdx and 02cosxdx.Explain how the notation for Riemann sums, k=1nf(xk)x, corresponds to the notation for the definite integral, abf(x)dx.Give a geometrical explanation of why aaf(x)dx=0.Use Table 5.4 to rewrite 16(2x34x)dx as the difference of two integrals.Use geometry to find a formula for 0axdx, in terms of a.If f is continuous on [a, b] and abf(x)dx=0, what can you conclude about f?Approximating net area The following functions are negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. 11. f(x) = 2x 1 on [0, 4]Approximating net area The following functions are negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. 12. f(x) = 4 x3 on [3, 7]Approximating net area The following functions are negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. 13. f(x) = sin 2x on [/2, ]Approximating net area The following functions are negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. 14. f(x) = x3 1 on [2, 0]Approximating net area The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. c. Use the sketch in part (a) to show which intervals of [a, b] make positive and negative contributions to the net area. 15. f(x) = 4 2x on [0, 4]Approximating net area The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. c. Use the sketch in part (a) to show which intervals of [a, b] make positive and negative contributions to the net area. 16. f(x) = 8 2x2 on [0, 4]Approximating net area The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. c. Use the sketch in part (a) to show which intervals of [a, b] make positive and negative contributions to the net area. 17. f(x) = sin 2x on [0, 3/4]Approximating net area The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. c. Use the sketch in part (a) to show which intervals of [a, b] make positive and negative contributions to the net area. 18. f(x) = x3 on [1, 2]Approximating net area The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. c. Use the sketch in part (a) to show which intervals of [a, b] make positive and negative contributions to the net area. 19. f(x) = tan1 (3x 1) on [0, 1]Approximating net area The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. c. Use the sketch in part (a) to show which intervals of [a, b] make positive and negative contributions to the net area. 20. f(x) = xex on [1, 1]27E28E29E30EApproximating definite integrals Complete the following steps for the given integral and the given value of n. a. Sketch the graph of the integrand on the interval of integration. b. Calculate x and the grid points x0, x1, , xn, assuming a regular partition. c. Calculate the left and right Riemann sums for the given value of n. d. Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral. 55. 36(12x)dx; n = 6Approximating definite integrals Complete the following steps for the given integral and the given value of n. a. Sketch the graph of the integrand on the interval of integration. b. Calculate x and the grid points x0, x1, , xn, assuming a regular partition. c. Calculate the left and right Riemann sums for the given value of n. d. Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral. 54. 02(x22)dx; n = 4Approximating definite integrals Complete the following steps for the given integral and the given value of n. a. Sketch the graph of the integrand on the interval of integration. b. Calculate x and the grid points x0, x1, , xn, assuming a regular partition. c. Calculate the left and right Riemann sums for the given value of n. d. Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral. 57. 171xdx; n = 6Approximating definite integrals Complete the following steps for the given integral and the given value of n. a. Sketch the graph of the integrand on the interval of integration. b. Calculate x and the grid points x0, x1, , xn, assuming a regular partition. c. Calculate the left and right Riemann sums for the given value of n. d. Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral. 56. 0/2cosxdx; n = 435E36EIdentifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function f on [a, b]. Identify f and express the limit as a definite integral. 23. lim0k=1nxx(lnxk)xkon [1, 2]38ENet area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. 25. 04(82x)dxNet area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. 26. 42(2x+4)dxNet area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. 27. 12(x)dxNet area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. 28. 02(1x)dxNet area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. 29. 0416x2dxNet area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. 30. 134(x1)2dxNet area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. 31. 04f(x)dx, where f(x)={53x1ifx2ifx2Net area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. 32. 110g(x)dx, where g(x)={4xif0x28x+16if2x38ifx3Net area from graphs The accompanying figure shows four regions bounded by the graph of y = x sin x: R1, R2, R3, and R4, whose areas are 1, 1, + 1, and 2 1, respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. 37. 0xsinxdxNet area from graphs The accompanying figure shows four regions bounded by the graph of y = x sin x: R1, R2, R3, and R4, whose areas are 1, 1, + 1, and 2 1, respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. 38. 03/2xsinxdxNet area from graphs The accompanying figure shows four regions bounded by the graph of y = x sin x: R1, R2, R3, and R4, whose areas are 1, 1, + 1, and 2 1, respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. 39. 02xsinxdxNet area from graphs The accompanying figure shows four regions bounded by the graph of y = x sin x: R1, R2, R3, and R4, whose areas are 1, 1, + 1, and 2 1, respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. 40. /22xsinxdxProperties of integrals Use only the fact that 043x(4x)dx=32 and the definitions and properties of integrals to evaluate the following integrals, if possible. a. 403x(4x)dx b. 04x(x4)dx c. 406x(4x)dx d. 083x(4x)dxProperties of integrals Suppose 14f(x)dx=8 and 16f(x)dx=5. Evaluate the following integrals. a. 14(3f(x))dx b. 143f(x)dx c. 6412f(x)dx d. 463f(x)dxProperties of integrals Suppose 03f(x)dx=2, 36f(x)dx=5, and 36g(x)dx=1. Evaluate the following integrals. a. 035f(x)dx b. 36(3g(x))dx c. 36(3f(x)g(x))dx d. 63(f(x)+2g(x))dxProperties of integrals Suppose f(x) 0 on [0, 2], f(x) 0 on [2, 5], 02f(x)dx=6, and 25f(x)dx=8. Evaluate the following integrals. a. 05f(x)dx b. 05f(x)dx c. 254f(x)dx d. 05(f(x)f(x))dxMore properties of integrals Consider two functions f and g on [1, 6] such that 16f(x)dx=10, 16g(x)dx=5, 46f(x)dx=5, and 14g(x)dx=2. Evaluate the following integrals. a. 143f(x)dx b. 16(f(x)g(x))dx c. 14(f(x)g(x))dx d. 46(g(x)f(x))dx e. 468g(x)dx f. 412f(x)dxSuppose f is continuous on [1, 5] and 2 f(x) 3 for all x in [1, 5]. Find lower and upper bounds for 15f(x)dx.Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 45. I=01(x32x)dx=34 a. 01(4x2x3)dx b. 10(2xx3)dxUsing properties of integrals Use the value of the first integral I to evaluate the two given integrals. 46. I=0/2(cos2sin)d=1 a. 0/2(2sincos)d b. /20(4cos8sin)dNet area from graphs The figure shows the areas of regions bounded by the graph of f and the x-axis. Evaluate the following integrals. 33. 0af(x)dxNet area from graphs The figure shows the areas of regions bounded by the graph of f and the x-axis. Evaluate the following integrals. 34. 0bf(x)dxNet area from graphs The figure shows the areas of regions bounded by the graph of f and the x-axis. Evaluate the following integrals. 35. acf(x)dxNet area from graphs The figure shows the areas of regions bounded by the graph of f and the x-axis. Evaluate the following integrals. 36. 0cf(x)dxDefinite integrals from graphs The figure shows the areas of regions bounded by the graph of f and the x-axis. Evaluate the following integrals. 63.0c|f(x)|dxDefinite integrals from graphs The figure shows the areas of regions bounded by the graph of f and the x-axis. Evaluate the following integrals. 64.0c(2|f(x)|+3f(x))dxDefinite integrals from graphs The figure shows the areas of regions bounded by the graph of f and the x-axis. Evaluate the following integrals. 65.a0f(x)dxDefinite integrals from graphs The figure shows the areas of regions bounded by the graph of f and the x-axis. Evaluate the following integrals. 66.c0|f(x)|dxUse geometry and properties of integrals to evaluate 01(2x+1x2+1)dx.Use geometry and properties of integrals to evaluate 15(|x2|+x2+6x5)dx.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If f is a constant function on the interval [a, b], then the right and left Riemann sums give the exact value of abf(x)dx. for any positive integer n. b. If f is a linear function on the interval [a, b], then a midpoint Riemann sum gives the exact value of abf(x)dx, for any positive integer n. c. 02sinaxdx=02/acosaxdx=0. (Hint: Graph the functions and use properties of trigonometric functions.) d. If abf(x)dx=baf(x)dx, then f is a constant function. e. Property 4 of Table 5.4 implies that abxf(x)dx=xabf(x)dx.Approximating definite integrals with a calculator Consider the following definite integrals. a.Write the left and right Riemann sums in sigma notation for an arbitrary value of n. b.Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. 70.493xdxApproximating definite integrals with a calculator Consider the following definite integrals. a.Write the left and right Riemann sums in sigma notation for an arbitrary value of n. b.Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. 71.01(x2+1)dxApproximating definite integrals with a calculator Consider the following definite integrals. a.Write the left and right Riemann sums in sigma notation for an arbitrary value of n. b.Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. 72.1elnxdxApproximating definite integrals with a calculator Consider the following definite integrals. a.Write the left and right Riemann sums in sigma notation for an arbitrary value of n. b.Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. 73.01cos1xdxApproximating definite integrals with a calculator Consider the following definite integrals. a.Write the left and right Riemann sums in sigma notation for an arbitrary value of n. b.Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. 74.11cos(x2)dxMidpoint Riemann sums with a calculator Consider the following definite integrals. a. Write the midpoint Riemann sum in sigma notation for an arbitrary value of n. b. Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. 63. 142xdxMidpoint Riemann sums with a calculator Consider the following definite integrals. a. Write the midpoint Riemann sum in sigma notation for an arbitrary value of n. b. Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. 64. 12sin(x4)dxMidpoint Riemann sums with a calculator Consider the following definite integrals. a. Write the midpoint Riemann sum in sigma notation for an arbitrary value of n. b. Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. 65. 04(4xx2)dxMidpoint Riemann sums with a calculator Consider the following definite integrals. a.Write the midpoint Riemann sum in sigma notation for an arbitrary value of n. b.Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. 78.011(sin1x+1)dxLimits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. 47. 02(2x+1)dxLimits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. 48. 15(1x)dxLimits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. 49. 37(4x+6)dxLimits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. 50. 02(x21)dxLimits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. 51. 14(x21)dxLimits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. 84.02(x3+x+1)dxLimits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. 85.01(4x3+3x2)dxArea by geometry Use geometry to evaluate the following integrals. 73. 162x4dxArea by geometry Use geometry to evaluate the following integrals. 75. 64242xx2dxIntegrating piecewise continuous functions Suppose f is continuous on the intervals [a, p] and [p, b], where a p b, with a finite jump at p. Form a uniform partition on the interval [a, p] with n grid points and another uniform partition on the interval [p, b] with m grid points, where p is a grid point of both partitions. Write a Riemann sum for abf(x)dx and separate it into two pieces for [a, p] and [p, b]. Explain why abf(x)dx=apf(x)dx+pbf(x)dx.Integrating piecewise continuous functions Use geometry and the result of Exercise 88 to evaluate the following integrals. 89.010f(x)dx,wheref(x)={2if0x53if5x10Integrating piecewise continuous functions Use geometry and the result of Exercise 88 to evaluate the following integrals. 90.16f(x)dx,wheref(x)={2xif1x4102xif4x6Integrating piecewise continuous functions Recall that the floor function x is the greatest integer less than or equal to x and that the ceiling function x is the least integer greater than or equal to x. Use the result of Exercise 88 and the graphs to evaluate the following integrals. 91.15xxdxIntegrating piecewise continuous functions Recall that the floor function x is the greatest integer less than or equal to x and that the ceiling function x is the least integer greater than or equal to x. Use the result of Exercise 88 and the graphs to evaluate the following integrals. 92.04xxdxConstants in integrals Use the definition of the definite integral to justify the property abcf(x)dx=cabf(x)dx, where f is continuous and c is a real number.Zero net area If 0 c d, then find the value of b (in terms of c and d) for which cd(x+b)dx=0.A nonintegrable function Consider the function defined on [0, 1] such that f(x) 1 if x is a rational number and f(x) = 0 if x is irrational. This function has an infinite number of discontinuities, and the integral 01f(x)dx does not exist. Show that the right, left, and midpoint Riemann sums on regular partitions with n subintervals equal 1 for all n. (Hint: Between any two real numbers lie a rational and an irrational number.)Powers of x by Riemann sums Consider the integral I(p) = 01xpdx, where p is a positive integer. a. Write the left Riemann sum for the integral with n subintervals. b. It is a fact (proved by the 17th-century mathematicians Fermat and Pascal) that limn1nk=0n1(kn)p=1p+1. Use this fact to evaluate I(p).An exact integration formula Evaluate abdxx2, where 0 a b, using the definition of the definite integral and the following steps. a. Assume {x0, x1, , xn} is a partition of [a, b] with xk = xk xk1, for k = 1, 2, , n. Show that xk1xk1xkxk, for k = 1, 2, , n. b. Show that 1xk11xk=xkxk1xk, for k = 1, 2, , n. c. Simplify the general Riemann sum for abdxx2 using xk=xk1xk. d. Conclude that abdxx2=1a1b.Use Property 3 of Table 5.4 and Property 7 of Table 5.5 to prove Property 8 of Table 5.5.In Example 1, let B(x) be the area function for f with left endpoint 5. Evaluate B(5) and B(9). Example 1 Comparing Area Functions The graph of f is shown in Figure 5.37 with areas of various regions marked. Let A(x)=1xf(t)dt and F(x)=3xf(t)dt be two area functions for f (note the different left endpoints). Evaluate the following area functions.Verify that the area function in Example 2c gives the correct area when x = 6 and x = 10. Example 2 Area of a Trapezoid Consider the trapezoid bounded by the line f(t)=2t+3 and the t-axis from t = 2 to t = x (Figure 5.38). The area function A(x)=2xf(t)dt gives the area of the trapezoid, for x 2. a.Evaluate A(2). b.Evaluate A(5). c.Find and graph the area function y = A(x), for x 2. d.Compare the derivative of A to f.Evaluate (xx+1)|12.Explain why f is an antiderivative of f.Suppose A is an area function of f. What is the relationship between f and A?Suppose F is an antiderivative of f and A is an area function of f. What is the relationship between F and A?Explain in words and write mathematically how the Fundamental Theorem of Calculus is used to evaluate definite integrals.Let f(x) = c, where c is a positive constant. Explain why an area function of f is an increasing function.The linear function f(x) = 3 x is decreasing on the interval [0, 3]. Is the area function for f (with left endpoint 0) increasing or decreasing on the interval [0, 3]? Draw a picture and explain.Evaluate 023x2dx and 223x2dx.Explain in words and express mathematically the inverse relationship between differentiation and integration as given by Part 1 of the Fundamental Theorem of Calculus.Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?Evaluate ddxaxf(t)dt and ddxabf(t)dt, where a and b are constants.Explain why abf(x)dx=f(b)f(a).Evaluate 38f(t)dt, where f is continuous on [3, 8], f(3) = 4, and f(8) = 20.Evaluate 273dx using the Fundamental Theorem of Calculus. Check your work by evaluating the integral using geometry.13EArea functions The graph of f is shown in the figure. Let A(x)=0xf(t)dt and F(x)=2xf(t)dt be two area functions for f. Evaluate the following area functions. a. A(2) b. F(5) c. A(0) d. F(8) e. A(8) f. A(5) g. F(2)Area functions for constant functions Consider the following functions f and real numbers a (see figure). a. Find and graph the area function A(x)=axf(t)dt for f. b. Verify that A(x) = f(x). 13. f(t) = 5, a = 0Area functions for constant functions Consider the following functions f and real numbers a (see figure). a. Find and graph the area function A(x)=axf(t)dt for f. b. Verify that A(x) = f(x). 14. f(t) = 10, a = 4Area functions for the same linear function Let f(t) = t and consider the two area functions A(x)=0xf(t)dt and F(x)=2xf(t)dt. a. Evaluate A(2) and A(4). Then use geometry to find an expression for A(x), for x 0. b. Evaluate F(4) and F(6). Then use geometry to find an expression for F(x), for x 2. c. Show that A(x) F(x) is a constant and that A(x) = F(x) = f(x).Area functions for the same linear function Let f(t) = 2t 2 and consider the two area functions A(x)=1xf(t)dt and F(x)=4xf(t)dt. a. Evaluate A(2) and A(3). Then use geometry to find an expression for A(x), for x 1. b. Evaluate F(5) and F(6). Then use geometry to find an expression for F(x), for x 4. c. Show that A(x) F(x) is a constant and that A(x) = F(x) = f(x).Area functions for linear functions Consider the following functions f and real numbers a (see figure). a. Find and graph the area function A(x)=axf(t)dt. b. Verify that A(x) = f(x). 19. f(t) = t + 5, a = 5Area functions for linear functions Consider the following functions f and real numbers a (see figure). a. Find and graph the area function A(x)=axf(t)dt. b. Verify that A(x) = f(x). 20. f(t) = 2t + 5, a = 0Area functions for linear functions Consider the following functions f and real numbers a (see figure). a. Find and graph the area function A(x)=axf(t)dt. b. Verify that A(x) = f(x). 21. f(t) = 3t + 1, a = 2Area functions for linear functions Consider the following functions f and real numbers a (see figure). a. Find and graph the area function A(x)=axf(t)dt. b. Verify that A(x) = f(x). 22. f(t) = 4t + 2, a = 0Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Explain why your result is consistent with the figure. 23. 01(x22x+3)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Explain why your result is consistent with the figure. 24. /47/4(sinx+cosx)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found. 25. 23(x2x6)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found. 26. 01(xx)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found. 27. 05(x29)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found. 28. 1/22(11x2)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 29. 024x3dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 30. 02(3x2+2x)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 31.188x1/3dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 32.116x5/4dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 31. 01(x+x)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 32. 0/42cosxdxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 33. 192xdxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 34. 492+ttdtDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 35. 22(x24)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 36. 0ln8exdxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 39.1/21(t38)dtDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 40.04t(t2)(t4)dtDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 43. 14(1x)(x4)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 40. 01/2dx1x2Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 41. 21x3dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 42. 0(1sinx)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 39. 0/4sec2dDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 44. /2/2(cosx1)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 45. 123tdtDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 48.49xxx2dxDefinite integrals Evaluate the following definite integrals using the Fundamental Theorem of Calculus. 91. 18y3dyDefinite integrals Evaluate the following definite integrals using the Fundamental Theorem of Calculus. 86. 120ln2exdxDefinite integrals Evaluate the following definite integrals using the Fundamental Theorem of Calculus. 87. 14x2xdxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 52.122s24s3dsDefinite integrals Evaluate the following definite integrals using the Fundamental Theorem of Calculus. 89. 0/3secxtanxdxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 54./4/2csc2dDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 55./43/4(cot2x+1)dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 56.0110ex+3dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 57.1311+x2dxDefinite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 58.0/4secx(secx+cosx)dxDefinite integrals Evaluate the following definite integrals using the Fundamental Theorem of Calculus. 93. 12z2+4zdzDefinite integrals Evaluate the following definite integrals using the Fundamental Theorem of Calculus. 92. 22dxxx21Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 61.0f(x)dx,wheref(x)={sinx+1ifx/22cosx+2ifx/2Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 62.13g(x)dx,whereg(x)={3x2+4x+1ifx22x+5ifx2Areas Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question. 51. The region bounded by y = x1/2 and the x-axis between x = 1 and x = 4Areas Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question. 52. The region above the x-axis bounded by y = 4 x2Areas Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question. 53. The region below the x-axis bounded by y = x4 16Areas Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question. 54. The region bounded by y = 6 cos x and the x-axis between x = /2 and x =Areas of regions Find the area of the region bounded by the graph of f and the x-axis on the given interval. 55. f(x) = x2 25 on [2, 4]Areas of regions Find the area of the region bounded by the graph of f and the x-axis on the given interval. 56. f(x) = x3 1 on [1, 2]Areas of regions Find the area of the region bounded by the graph of f and the x-axis on the given interval. 57. f(x)=1xon[2,1]Areas of regions Find the area of the region bounded by the graph of f and the x-axis on the given interval. 58. f(x) = x(x + l)(x 2) on [1, 2]Areas of regions Find the area of the region bounded by the graph of f and the x-axis on the given interval. 59. f(x) = sin x on [/4, 3/4]Areas of regions Find the area of the region bounded by the graph of f and the x-axis on the given interval. 60. f(x) = cos x on [/2, ]Derivatives of integrals Simplify the following expressions. 61. ddx3x(t2+t+1)dtDerivatives and integrals Simplify the given expressions. 102. ddxx1et2dtDerivatives of integrals Simplify the following expressions. 65. ddxx1t4+1dtDerivatives of integrals Simplify the following expressions. 66. ddxx0dpp2+1Derivatives of integrals Simplify the following expressions. 63. ddx2x3dpp2Derivatives and integrals Simplify the given expressions. 100. ddx0x2dtt2+4Derivatives and integrals Simplify the given expressions. 101. ddx0cosx(t4+6)dtDerivatives of integrals Simplify the following expressions. 80.ddw0wln(x2+1)dxDerivatives of integrals Simplify the following expressions. 81.ddzsinz10dtt4+1Derivatives of integrals Simplify the following expressions. 82.ddyy310x6+1dxDerivatives and integrals Simplify the given expressions. 103. ddt(1t3xdxt213xdx)Derivatives and integrals Simplify the given expressions. 104. ddt(0tdx1+x2+01/tdx1+x2)Derivatives of integrals Simplify the following expressions. 85.ddx0x1+t2dtDerivatives of integrals Simplify the following expressions. 68. ddxete2nlnt2dt87EWorking with area functions Consider the function f and its graph. a. Estimate the zeros of the area function A(x)=0xf(t)dt, for 0 x 10. b. Estimate the points (if any) at which A has a local maximum or minimum. c. Sketch a graph of A, for 0 x 10, without a scale on the y-axis. 70.89E90E91E92EArea functions from graphs The graph of f is given in the figure. Let A(x)=0xf(t)dt and evaluate A(2), A(5), A(8), and A(12).94EWorking with area functions Consider the function f and the points a, b, and c. a. Find the area function A(x)=axf(t)dt using the Fundamental Theorem. b. Graph f and A. c. Evaluate A(b) and A(c). Interpret the results using the graphs of pan (b). 77. f(x) = ex; a = 0, b = ln 2, c = ln 4Working with area functions Consider the function f and the points a, b, and c. a. Find the area function A(x)=axf(t)dt using the Fundamental Theorem. b. Graph f and A. c. Evaluate A(b) and A(c). Interpret the results using the graphs of part (b). 78. f(x) = 12x(x l)(x 2); a = 0, b = l, c = 297E98EFind the critical points of the function f(x)=1xt2(t3)(t4)dt, and determine the intervals on which f is increasing or decreasing.Determine the intervals on which the function g(x)=x0tt2+1dt is concave up or concave down.101E102EAreas of regions Find the area of the region R bounded by the graph of f and the x-axis on the given interval. Graph f and show the region R. 95. f(x) = 2 |x| on [2, 4]104EAreas of regions Find the area of the region R bounded by the graph of f and the x-axis on the given interval. Graph f and show the region R. 97. f(x) = x4 4 on [l, 4]Areas of regions Find the area of the region R bounded by the graph of f and the x-axis on the given interval. Graph f and show the region R. 98. f(x) = x2(x 2) on [1, 3]Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.Suppose f is a positive decreasing function, for x 0. Then the area function A(x)=0xf(t)dt is an increasing function of x. b.Suppose f is a negative increasing function, for x 0. Then the area function A(x)=0xf(t)dt is a decreasing function of x. c.The functions p(x) = sin 3x and q(x) = 4 sin 3x are antiderivatives of the same function. d.If A(x) = 3x2 x 3 is an area function for f, then B(x) = 3x2 x is also an area function for f.Explorations and Challenges Evaluate limx22xt2+t+3dtx24.Maximum net area What value of b 1 maximizes the integral 1bx2(3x)dx?Maximum net area Graph the function f(x) = 8 + 2x x2 and determine the values of a and b that maximize the value of the integral ab(8+2xx2)dx.111ECubic zero net area Consider the graph of the cubic y = x(x a)(x b), where 0 a b. Verify that the graph bounds a region above the x-axis, for 0 x a, and bounds a region below the x-axis, for a x b. What is the relationship between a and b if the areas of these two regions are equal?An integral equation Use the Fundamental Theorem of Calculus, Part 1, to find the function f that satisfies the equation 0xf(t)dt=2cosx+3x2. Verify the result by substitution into the equation.114EAsymptote of sine integral Use a calculator to approximate limxS(x)=limx0xsinttdt, where S is the sine integral function (see Example 7). Explain your reasoning.Sine integral Show that the sine integral S(x)=0xsinttdt satisfies the (differential) equation xS(x) + 2S(x) + xS(x) = 0.117EContinuity at the endpoints Assume that f is continuous on [a, b] and let A be the area function for f with left endpoint a. Let m and M be the absolute minimum and maximum values of f on [a, b], respectively. a. Prove that m(x a) A(x) M(x a) for all x in [a, b]. Use this result and the Squeeze Theorem to show that A is continuous from the right at x = a. b. Prove that m(b x) A(b) A(x) M(b x) for all x in [a, b]. Use this result to show that A is continuous from the left at x = b.Discrete version of the Fundamental Theorem In this exercise, we work with a discrete problem and show why the relationship abf(x)dx=f(b)f(a)makes sense. Suppose we have a set of equally spaced grid points a=x0x1x2xn1xn=b, where the distance between any two grid points is x. Suppose also that at each grid point xk, a function value f(xk) is defined, for k = 0, , n. a. We now replace the integral with a sum and replace the derivative with a difference quotient. Explain why abf(x)dx is analogous to k=1nf(xk)f(xk1)x=f(xk)x. b. Simplify the sum in part (a) and show that it is equal to f(b) f(a). c. Explain the correspondence between the integral relationship and the summation relationship.If f and g are both even functions, is the product fg even or odd? Use the facts that f(x)=f(x) and g(x)=g(x).2QCExplain why f(x) = 0 for at least one point of (a, b) if f is continuous and baf(x)dx=0.If f is an odd function, why is aaf(x)dx=0?If f is an even function, why is aaf(x)dx=20af(x)dx?Using symmetry Suppose f is an even function and 88f(x)dx=18. a. Evaluate 08f(x)dx b. Evaluate 88xf(x)dxUsing symmetry Suppose f is an odd function, 04f(x)dx=3, and 08f(x)dx=9. a. Evaluate 48f(x)dx b. Evaluate 84f(x)dxUse symmetry to explain why 44(5x4+3x3+2x2+x+1)dx=204(5x4+2x2+1)dx.Use symmetry to fill in the blanks: (sinx+cosx)dx=0dx.Is x12 an even or odd function? Is sin x2 an even or odd function?8E9E10ESymmetry in integrals Use symmetry to evaluate the following integrals. 7. 22x9dxSymmetry in integrals Use symmetry to evaluate the following integrals. 8. 2002002x5dxSymmetry in integrals Use symmetry to evaluate the following integrals. 9. 22(3x82)dxSymmetry in integrals Use symmetry to evaluate the following integrals. 10. /4/4cosxdxSymmetry in integrals Use symmetry to evaluate the following integrals. 15.22(x2+x3)dxSymmetry in integrals Use symmetry to evaluate the following integrals. 16.t2sintdtSymmetry in integrals Use symmetry to evaluate the following integrals. 11. 22(x93x5+2x210)dxSymmetry in integrals Use symmetry to evaluate the following integrals. 18./2/25sindSymmetry in integrals Use symmetry to evaluate the following integrals. 19./4/4sin5tdtSymmetry in integrals Use symmetry to evaluate the following integrals. 16. 11(1|x|)dxSymmetry in integrals Use symmetry to evaluate the following integrals. 43. /4/4sec2xdxSymmetry in integrals Use symmetry to evaluate the following integrals. 22./4/4tandSymmetry in integrals Use symmetry to evaluate the following integrals. 45. 22x34xx2+1dxSymmetry in integrals Use symmetry to evaluate the following integrals. 44. 22(1|x|3)dxAverage values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value. 21. f(x) = x3 on [l, l]Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value. 22. f(x) = x2 + 1 on [2, 2]Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value. 23. f(x)=1x2+1on[1,1]Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value. 25. f(x) = l/x on [1, e]Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value. 27. f(x)=cosxon[2,2]Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value. 28. f(x) = x(1 x) on [0, 1]Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value. 29. f(x) = xn on [0, 1], for any positive integer nAverage values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value. 30. f(x) = x1/n on [0, 1], for any positive integer nAverage distance on a parabola What is the average distance between the parabola y = 30x(20 x) and the x-axis on the interval [0, 20]?Average elevation The elevation of a path is given by f(x) = x3 5x2 + 30, where x measures horizontal distances. Draw a graph of the elevation function and find its average value, for 0 x 4.Average velocity The velocity in m/s of an object moving along a line over the time interval [0, 6] is v(t)=t2+3t. Find the average velocity of the object over this time interval.Average velocity A rock is launched vertically upward from the ground with a speed of 64 ft/s. The height of the rock (in ft) above the ground after t seconds is given by the function s(t)=16t2+64t. Find its average velocity during its flight.Average height of an arch The height of an arch above the ground is given by the function y = 10 sin x, for 0 x . what is the average height of the arch above the ground?Average height of a wave The surface of a water wave is described by y = 5(1 + cos x), for x , where y = 0 corresponds to a trough of the wave (see figure). Find the average height of the wave above the trough on [, ].Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval. 35. f(x) = 8 2x on [0, 4]Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval. 36. f(x) = ex on [0, 2]Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval. 37. f(x) = 1 x2/a2 on [0, a], where a is a positive real numberMean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval. 38. f(x)=4sinxon[0,]Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval. 39. f(x) = l |x| on [l, 1]Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval. 40. f(x) = 1/x on [1, 4]Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If f is symmetric about the line x = 2, then 04f(x)dx=202f(x)dx. b. If f has the property f(a + x) = f(a x), for all x, where a is a constant, then a2a+2f(x)dx=0. c. The average value of a linear function on an interval [a, b] is the function value at the midpoint of [a, b]. d. Consider the function f(x) = x(a x) on the interval [0, a], for a 0. Its average value on [0, a] is 12 of its maximum value.Planetary orbits The planets orbit the Sun in elliptical orbits with the Sun at one focus (see Section 12.4 for more on ellipses). The equation of an ellipse whose dimensions are 2a in the x-direction and 2b in the y-direction is .
Let d2 denote the square of the distance from a planet to the center of the ellipse at (0, 0). Integrate over the interval [−a, a] to show that the average value of d2 is .
Show that in the case of a circle (a = b = R), the average value in part (a) is R2.
Assuming 0 < b < a, the coordinates of the Sun are (, 0). Let D2 denote the square of the distance from the planet to the Sun. Integrate over the interval [−a, a] to show that the average value of D2 is .
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 parabola y=630(1(x315)2). Find the average height of the arch above the ground.