Problem Set: The Definite Integral

In the following exercises, express the limits as integrals.

1. [latex]\underset{n\to \infty }{\lim}\underset{i=1}{\overset{n}{\Sigma}}(x_i^*) \Delta x[/latex] over [latex][1,3][/latex]

2. [latex]\underset{n\to \infty }{\lim}\underset{i=1}{\overset{n}{\Sigma}}(5(x_i^*)^2-3(x_i^*)^3) \Delta x[/latex] over [latex][0,2][/latex]

3. [latex]\underset{n\to \infty }{\lim}\underset{i=1}{\overset{n}{\Sigma}} \sin^2 (2\pi x_i^*) \Delta x[/latex] over [latex][0,1][/latex]

4. [latex]\underset{n\to \infty }{\lim}\underset{i=1}{\overset{n}{\Sigma}} \cos^2 (2\pi x_i^*) \Delta x[/latex] over [latex][0,1][/latex]

In the following exercises, given [latex]L_n[/latex] or [latex]R_n[/latex] as indicated, express their limits as [latex]n\to \infty[/latex] as definite integrals, identifying the correct intervals.

5. [latex]L_n=\frac{1}{n}\underset{i=1}{\overset{n}{\Sigma}}\frac{i-1}{n}[/latex]

6. [latex]R_n=\frac{1}{n}\underset{i=1}{\overset{n}{\Sigma}}\frac{i}{n}[/latex]

7. [latex]L_n=\frac{2}{n}\underset{i=1}{\overset{n}{\Sigma}}(1+2\frac{i-1}{n})[/latex]

8. [latex]R_n=\frac{3}{n}\underset{i=1}{\overset{n}{\Sigma}}(3+3\frac{i}{n})[/latex]

9. [latex]L_n=\frac{2\pi }{n}\underset{i=1}{\overset{n}{\Sigma}}2\pi \frac{i-1}{n} \cos (2\pi \frac{i-1}{n})[/latex]

10. [latex]R_n=\frac{1}{n}\underset{i=1}{\overset{n}{\Sigma}}(1+\frac{i}{n})\log((1+\frac{i}{n})^2)[/latex]

In the following exercises (11-16), evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the [latex]x[/latex]-axis.

11. A graph containing the upper half of three circles on the x axis. The first has center at (1,0) and radius one. It corresponds to the function sqrt(2x – x^2) over [0,2]. The second has center at (4,0) and radius two. It corresponds to the function sqrt(-12 + 8x – x^2) over [2,6]. The last has center at (9,0) and radius three. It corresponds to the function sqrt(-72 + 18x – x^2) over [6,12]. All three semi circles are shaded – the area under the curve and above the x axis.

12. A graph of three isosceles triangles corresponding to the functions 1 - |x-1| over [0,2], 2 - |x-4| over [2,4], and 3 - |x-9| over [6,12]. The first triangle has endpoints at (0,0), (2,0), and (1,1). The second triangle has endpoints at (2,0), (6,0), and (4,2). The last has endpoints at (6,0), (12,0), and (9,3). All three are shaded.

13. A graph with three parts. The first is the upper half of a circle with center at (1, 0) and radius 1, which corresponds to the function sqrt(2x – x^2) over [0,2]. The second is a triangle with endpoints at (2, 0), (6, 0), and (4, -2), which corresponds to the function |x-4| - 2 over [2, 6]. The last is the upper half of a circle with center at (9, 0) and radius 3, which corresponds to the function sqrt(-72 + 18x – x^2) over [6,12]. All three are shaded.
14. A graph of three shaded triangles. The first has endpoints at (0, 0), (2, 0), and (1, 1) and corresponds to the function 1 - |x-1| over [0, 2]. The second has endpoints at (2, 0), (6, 0), and (4, -2) and corresponds to the function |x-4| - 2 over [2, 6]. The third has endpoints at (6, 0), (12, 0), and (9, 3) and corresponds to the function 3 - |x-9| over [6, 12].

15. A graph with three shaded parts. The first is the upper half of a circle with center at (1, 0) and radius one. It corresponds to the function sqrt(2x – x^2) over [0, 2]. The second is the lower half of a circle with center at (4, 0) and radius two, which corresponds to the function -sqrt(-12 + 8x – x^2) over [2, 6]. The last is the upper half of a circle with center at (9, 0) and radius three. It corresponds to the function sqrt(-72 + 18x – x^2) over [6, 12].

16. A graph with three shaded parts. The first is a triangle with endpoints at (0, 0), (2, 0), and (1, 1), which corresponds to the function 1 - |x-1| over [0, 2] in quadrant 1. The second is the lower half of a circle with center at (4, 0) and radius two, which corresponds to the function –sqrt(-12 + 8x – x^2) over [2, 6]. The last is a triangle with endpoints at (6, 0), (12, 0), and (9, 3), which corresponds to the function 3 - |x-9| over [6, 12].

In the following exercises (17-24), evaluate the integral using area formulas.

17. [latex]\displaystyle\int_0^3 (3-x) dx[/latex]

18. [latex]\displaystyle\int_2^3 (3-x) dx[/latex]

19. [latex]\displaystyle\int_{-3}^3 (3-|x|) dx[/latex]

20. [latex]\displaystyle\int_0^6 (3-|x-3|) dx[/latex]

21. [latex]\displaystyle\int_{-2}^2 \sqrt{4-x^2} dx[/latex]

22. [latex]\displaystyle\int_1^5 \sqrt{4-(x-3)^2} dx[/latex]

23. [latex]\displaystyle\int_0^{12} \sqrt{36-(x-6)^2} dx[/latex]

24. [latex]\displaystyle\int_{-2}^3 (3-|x|) dx[/latex]

In the following exercises (25-28), use averages of values at the left ([latex]L[/latex]) and right ([latex]R[/latex]) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals.

25. [latex]\{(0,0),(2,1),(4,3),(5,0),(6,0),(8,3)\}[/latex] over [latex][0,8][/latex]

26. [latex]\{(0,2),(1,0),(3,5),(5,5),(6,2),(8,0)\}[/latex] over [latex][0,8][/latex]

27. [latex]\{(-4,-4),(-2,0),(0,-2),(3,3),(4,3)\}[/latex] over [latex][-4,4][/latex]

28. [latex]\{(-4,0),(-2,2),(0,0),(1,2),(3,2),(4,0)\}[/latex] over [latex][-4,4][/latex]

Suppose that [latex]\displaystyle\int_0^4 f(x) dx=5[/latex] and [latex]\displaystyle\int_0^2 f(x) dx=-3[/latex], and [latex]\displaystyle\int_0^4 g(x) dx=-1[/latex] and [latex]\displaystyle\int_0^2 g(x) dx=2[/latex]. In the following exercises (29-34), compute the integrals.

29. [latex]\displaystyle\int_0^4 (f(x)+g(x)) dx[/latex]

30. [latex]\displaystyle\int_2^4 (f(x)+g(x)) dx[/latex]

31. [latex]\displaystyle\int_0^2 (f(x)-g(x)) dx[/latex]

32. [latex]\displaystyle\int_2^4 (f(x)-g(x)) dx[/latex]

33. [latex]\displaystyle\int_0^2 (3f(x)-4g(x)) dx[/latex]

34. [latex]\displaystyle\int_2^4 (4f(x)-3g(x)) dx[/latex]

In the following exercises (35-38), use the identity [latex]\displaystyle\int_{−A}^A f(x) dx = \displaystyle\int_{−A}^0 f(x) dx + \displaystyle\int_0^A f(x) dx[/latex] to compute the integrals.

35. [latex]\displaystyle\int_{−\pi}^{\pi} \frac{\sin t}{1+t^2} dt[/latex] (Hint: [latex]\sin(−t)=−\sin (t)[/latex])

36. [latex]\displaystyle\int_{−\sqrt{\pi}}^{\sqrt{\pi}} \frac{t}{1+ \cos t} dt[/latex]

37. [latex]\displaystyle\int_1^3 (2-x) dx[/latex] (Hint: Look at the graph of [latex]f[/latex].)

38. [latex]{\displaystyle\int }_{2}^{4}{(x-3)}^{3}dx[/latex] (Hint: Look at the graph of [latex]f[/latex].)

In the following exercises (39-44), given that [latex]\displaystyle\int_0^1 x dx = \frac{1}{2}, \, \displaystyle\int_0^1 x^2 dx = \frac{1}{3}[/latex], and [latex]\displaystyle\int_0^1 x^3 dx = \frac{1}{4}[/latex], compute the integrals.

39. [latex]\displaystyle\int_0^1 (1+x+x^2+x^3) dx[/latex]

40. [latex]\displaystyle\int_0^1 (1-x+x^2-x^3) dx[/latex]

41. [latex]\displaystyle\int_0^1 (1-x)^2 dx[/latex]

42. [latex]\displaystyle\int_0^1 (1-2x)^3 dx[/latex]

43. [latex]\displaystyle\int_0^1 (6x-\frac{4}{3}x^2) dx[/latex]

44. [latex]\displaystyle\int_0^1 (7-5x^3) dx[/latex]

In the following exercises (45-50), use the comparison theorem.

45. Show that [latex]\displaystyle\int_0^3 (x^2-6x+9) dx \ge 0[/latex].

46. Show that [latex]\displaystyle\int_{-2}^3 (x-3)(x+2) dx \le 0[/latex].

47. Show that [latex]\displaystyle\int_0^1 \sqrt{1+x^3} dx \le \displaystyle\int_0^1 \sqrt{1+x^2} dx[/latex].

48. Show that [latex]\displaystyle\int_1^2 \sqrt{1+x} dx \le \displaystyle\int_1^2 \sqrt{1+x^2} dx[/latex].

49. Show that [latex]\displaystyle\int_0^{\pi/2} \sin t dt \ge \frac{\pi}{4}[/latex]. (Hint: [latex]\sin t \ge \frac{2t}{\pi}[/latex] over [latex][0,\frac{\pi}{2}][/latex])

50. Show that [latex]\displaystyle\int_{−\pi/4}^{\pi/4} \cos t dt \ge \pi \sqrt{2}/4[/latex].

In the following exercises (51-56), find the average value [latex]f_{\text{ave}}[/latex] of [latex]f[/latex] between [latex]a[/latex] and [latex]b[/latex], and find a point [latex]c[/latex], where [latex]f(c)=f_{\text{ave}}[/latex].

51. [latex]f(x)=x^2, \, a=-1, \, b=1[/latex]

52. [latex]f(x)=x^5, \, a=-1, \, b=1[/latex]

53. [latex]f(x)=\sqrt{4-x^2}, \, a=0, \, b=2[/latex]

54. [latex]f(x)=(3-|x|), \, a=-3, \, b=3[/latex]

55. [latex]f(x)= \sin x, \, a=0, \, b=2\pi[/latex]

56. [latex]f(x)= \cos x, \, a=0, \, b=2\pi[/latex]

In the following exercises, approximate the average value using Riemann sums [latex]L_{100}[/latex] and [latex]R_{100}[/latex]. How does your answer compare with the exact given answer?

57. [T] [latex]y=\ln (x)[/latex] over the interval [latex][1,4][/latex]; the exact solution is [latex]\dfrac{\ln (256)}{3}-1[/latex].

58. [T] [latex]y=e^{x/2}[/latex] over the interval [latex][0,1][/latex]; the exact solution is [latex]2(\sqrt{e}-1)[/latex].

59. [T] [latex]y= \tan x[/latex] over the interval [latex][0,\frac{\pi}{4}][/latex]; the exact solution is [latex]\dfrac{2\ln (2)}{\pi}[/latex].

60. [T] [latex]y=\dfrac{x+1}{\sqrt{4-x^2}}[/latex] over the interval [latex][-1,1][/latex]; the exact solution is [latex]\frac{\pi }{6}[/latex].

In the following exercises, compute the average value using the left Riemann sums [latex]L_N[/latex] for [latex]N=1,10,100[/latex]. How does the accuracy compare with the given exact value?

61. [T] [latex]y=x^2-4[/latex] over the interval [latex][0,2][/latex]; the exact solution is [latex]-\frac{8}{3}[/latex].

62. [T] [latex]y=xe^{x^2}[/latex] over the interval [latex][0,2][/latex]; the exact solution is [latex]\frac{1}{4}(e^4-1)[/latex].

63. [T] [latex]y=\left(\frac{1}{2}\right)^x[/latex] over the interval [latex][0,4][/latex]; the exact solution is [latex]\dfrac{15}{64\ln (2)}[/latex].

64. [T] [latex]y=x \sin (x^2)[/latex] over the interval [latex][−\pi ,0][/latex]; the exact solution is [latex]\dfrac{\cos (\pi^2)-1}{2\pi}[/latex].

65. Suppose that [latex]A=\displaystyle\int_0^{2\pi} \sin^2 t dt[/latex] and [latex]B=\displaystyle\int_0^{2\pi} \cos^2 t dt[/latex]. Show that [latex]A+B=2\pi[/latex] and [latex]A=B[/latex].

66. Suppose that [latex]A=\displaystyle\int_{−\pi/4}^{\pi/4} \sec^2 t dt = \pi[/latex] and [latex]B=\displaystyle\int_{−\pi/4}^{\pi/4} \tan^2 t dt[/latex]. Show that [latex]B-A=\frac{\pi }{2}[/latex].

67. Show that the average value of [latex]\sin^2 t[/latex] over [latex][0,2\pi][/latex] is equal to [latex]\frac{1}{2}[/latex]. Without further calculation, determine whether the average value of [latex]\sin^2 t[/latex] over [latex][0,\pi][/latex] is also equal to [latex]\frac{1}{2}[/latex].

68. Show that the average value of [latex]\cos^2 t[/latex] over [latex][0,2\pi][/latex] is equal to [latex]1/2[/latex]. Without further calculation, determine whether the average value of [latex]\cos^2 (t)[/latex] over [latex][0,\pi][/latex] is also equal to [latex]1/2[/latex].

69. Explain why the graphs of a quadratic function (parabola) [latex]p(x)[/latex] and a linear function [latex]\ell (x)[/latex] can intersect in at most two points. Suppose that [latex]p(a)=\ell (a)[/latex] and [latex]p(b)=\ell (b)[/latex], and that [latex]\displaystyle\int_a^b p(t) dt > \displaystyle\int_a^b \ell (t) dt[/latex]. Explain why [latex]\displaystyle\int_c^d p(t) > \displaystyle\int_c^d \ell (t) dt[/latex] whenever [latex]a \le c < d \le b[/latex].

70. Suppose that parabola [latex]p(x)=ax^2+bx+c[/latex] opens downward [latex](a<0)[/latex] and has a vertex of [latex]y=\frac{−b}{2a}>0[/latex]. For which interval [latex][A,B][/latex] is [latex]\displaystyle\int_A^B (ax^2+bx+c) dx[/latex] as large as possible?

71. Suppose [latex][a,b][/latex] can be subdivided into subintervals [latex]a=a_0

  1. Explain why [latex]\displaystyle\int_a^b f(t) dt = A_1+A_2+ \cdots +A_N[/latex].
  2. Then, explain why [latex]|\displaystyle\int_a^b f(t) dt| \le \displaystyle\int_a^b |f(t)| dt[/latex].

72. Suppose [latex]f[/latex] and [latex]g[/latex] are continuous functions such that [latex]\displaystyle\int_c^d f(t) dt \le \displaystyle\int_c^d g(t) dt[/latex] for every subinterval [latex][c,d][/latex] of [latex][a,b][/latex]. Explain why [latex]f(x)\le g(x)[/latex] for all values of [latex]x[/latex].

73. Suppose the average value of [latex]f[/latex] over [latex][a,b][/latex] is 1 and the average value of [latex]f[/latex] over [latex][b,c][/latex] is 1 where [latex]a

74. Suppose that [latex][a,b][/latex] can be partitioned. taking [latex]a=a_0

75. Suppose that for each [latex]i[/latex] such that [latex]1\le i\le N[/latex] one has [latex]\displaystyle\int_{i-1}^i f(t) dt=i[/latex]. Show that [latex]\displaystyle\int_0^N f(t) dt=\frac{N(N+1)}{2}[/latex].

76. Suppose that for each [latex]i[/latex] such that [latex]1\le i\le N[/latex] one has [latex]\displaystyle\int_{i-1}^i f(t) dt=i^2[/latex]. Show that [latex]\displaystyle\int_0^N f(t) dt=\frac{N(N+1)(2N+1)}{6}[/latex].

77. [T] Compute the left and right Riemann sums [latex]L_{10}[/latex] and [latex]R_{10}[/latex] and their average [latex]\frac{L_{10}+R_{10}}{2}[/latex] for [latex]f(t)=t^2[/latex] over [latex][0,1][/latex]. Given that [latex]\displaystyle\int_0^1 t^2 dt=0.\bar{33}[/latex], to how many decimal places is [latex]\frac{L_{10}+R_{10}}{2}[/latex] accurate?

78. [T] Compute the left and right Riemann sums, [latex]L_{10}[/latex] and [latex]R_{10}[/latex], and their average [latex]\frac{L_{10}+R_{10}}{2}[/latex] for [latex]f(t)=(4-t^2)[/latex] over [latex][1,2][/latex]. Given that [latex]\displaystyle\int_1^2 (4-t^2) dt=1.\bar{66}[/latex], to how many decimal places is [latex]\frac{L_{10}+R_{10}}{2}[/latex] accurate?

79. If [latex]\displaystyle\int_1^5 \sqrt{1+t^4} dt=41.7133 \cdots[/latex], what is [latex]\displaystyle\int_1^5 \sqrt{1+u^4} du[/latex]?

80. Estimate [latex]\displaystyle\int_0^1 t dt[/latex] using the left and right endpoint sums, each with a single rectangle. How does the average of these left and right endpoint sums compare with the actual value [latex]\displaystyle\int_0^1 t dt[/latex]?

81. Estimate [latex]\displaystyle\int_0^1 t dt[/latex] by comparison with the area of a single rectangle with height equal to the value of [latex]t[/latex] at the midpoint [latex]t=\frac{1}{2}[/latex]. How does this midpoint estimate compare with the actual value [latex]\displaystyle\int_0^1 t dt[/latex]?

82. From the graph of [latex]\sin (2\pi x)[/latex] shown:

  1. Explain why [latex]\displaystyle\int_0^1 \sin (2\pi t) dt=0[/latex].
  2. Explain why, in general, [latex]\displaystyle\int_a^{a+1} \sin (2\pi t) dt=0[/latex] for any value of [latex]a[/latex].
    A graph of the function f(x) = sin(2pi*x) over [0, 2]. The function is shaded over [.7, 1] above the curve and below to x axis, over [1,1.5] under the curve and above the x axis, and over [1.5, 1.7] above the curve and under the x axis. The graph is antisymmetric with respect o t = ½ over [0,1].

83. If [latex]f[/latex] is 1-periodic [latex](f(t+1)=f(t))[/latex], odd, and integrable over [latex][0,1][/latex], is it always true that [latex]\displaystyle\int_0^1 f(t) dt=0[/latex]?

84. If [latex]f[/latex] is 1-periodic and [latex]\displaystyle\int_0^1 f(t) dt=A[/latex], is it necessarily true that [latex]\displaystyle\int_a^{1+a} f(t) dt=A[/latex] for all [latex]A[/latex]?