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.
12.
16.
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<a_1<a_2< \cdots <a_N=b[/latex] such that either [latex]f\ge 0[/latex] over [latex][a_{i-1},a_i][/latex] or [latex]f\le 0[/latex] over [latex][a_{i-1},a_i][/latex]. Set [latex]A_i=\displaystyle\int_{a_{i-1}}^{a_i} f(t) dt[/latex].
- Explain why [latex]\displaystyle\int_a^b f(t) dt = A_1+A_2+ \cdots +A_N[/latex].
- 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<c<b[/latex]. Show that the average value of [latex]f[/latex] over [latex][a,c][/latex] is also 1.
74. Suppose that [latex][a,b][/latex] can be partitioned. taking [latex]a=a_0<a_1< \cdots < a_N=b[/latex] such that the average value of [latex]f[/latex] over each subinterval [latex][a_{i-1},a_i]=1[/latex] is equal to 1 for each [latex]i=1\, \cdots , N[/latex]. Explain why the average value of [latex]f[/latex] over [latex][a,b][/latex] is also equal to 1.
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:
- Explain why [latex]\displaystyle\int_0^1 \sin (2\pi t) dt=0[/latex].
- Explain why, in general, [latex]\displaystyle\int_a^{a+1} \sin (2\pi t) dt=0[/latex] for any value of [latex]a[/latex].
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]?