## Problem Set: The Definite Integral

In the following exercises, express the limits as integrals.

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

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

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

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

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

5. $L_n=\frac{1}{n}\underset{i=1}{\overset{n}{\Sigma}}\frac{i-1}{n}$

6. $R_n=\frac{1}{n}\underset{i=1}{\overset{n}{\Sigma}}\frac{i}{n}$

7. $L_n=\frac{2}{n}\underset{i=1}{\overset{n}{\Sigma}}(1+2\frac{i-1}{n})$

8. $R_n=\frac{3}{n}\underset{i=1}{\overset{n}{\Sigma}}(3+3\frac{i}{n})$

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

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

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 $x$-axis.

11.

12.

13.
14.

15.

16.

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

17. $\displaystyle\int_0^3 (3-x) dx$

18. $\displaystyle\int_2^3 (3-x) dx$

19. $\displaystyle\int_{-3}^3 (3-|x|) dx$

20. $\displaystyle\int_0^6 (3-|x-3|) dx$

21. $\displaystyle\int_{-2}^2 \sqrt{4-x^2} dx$

22. $\displaystyle\int_1^5 \sqrt{4-(x-3)^2} dx$

23. $\displaystyle\int_0^{12} \sqrt{36-(x-6)^2} dx$

24. $\displaystyle\int_{-2}^3 (3-|x|) dx$

In the following exercises (25-28), use averages of values at the left ($L$) and right ($R$) 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. $\{(0,0),(2,1),(4,3),(5,0),(6,0),(8,3)\}$ over $[0,8]$

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

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

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

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

29. $\displaystyle\int_0^4 (f(x)+g(x)) dx$

30. $\displaystyle\int_2^4 (f(x)+g(x)) dx$

31. $\displaystyle\int_0^2 (f(x)-g(x)) dx$

32. $\displaystyle\int_2^4 (f(x)-g(x)) dx$

33. $\displaystyle\int_0^2 (3f(x)-4g(x)) dx$

34. $\displaystyle\int_2^4 (4f(x)-3g(x)) dx$

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

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

36. $\displaystyle\int_{−\sqrt{\pi}}^{\sqrt{\pi}} \frac{t}{1+ \cos t} dt$

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

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

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

39. $\displaystyle\int_0^1 (1+x+x^2+x^3) dx$

40. $\displaystyle\int_0^1 (1-x+x^2-x^3) dx$

41. $\displaystyle\int_0^1 (1-x)^2 dx$

42. $\displaystyle\int_0^1 (1-2x)^3 dx$

43. $\displaystyle\int_0^1 (6x-\frac{4}{3}x^2) dx$

44. $\displaystyle\int_0^1 (7-5x^3) dx$

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

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

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

47. Show that $\displaystyle\int_0^1 \sqrt{1+x^3} dx \le \displaystyle\int_0^1 \sqrt{1+x^2} dx$.

48. Show that $\displaystyle\int_1^2 \sqrt{1+x} dx \le \displaystyle\int_1^2 \sqrt{1+x^2} dx$.

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

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

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

51. $f(x)=x^2, \, a=-1, \, b=1$

52. $f(x)=x^5, \, a=-1, \, b=1$

53. $f(x)=\sqrt{4-x^2}, \, a=0, \, b=2$

54. $f(x)=(3-|x|), \, a=-3, \, b=3$

55. $f(x)= \sin x, \, a=0, \, b=2\pi$

56. $f(x)= \cos x, \, a=0, \, b=2\pi$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

71. Suppose $[a,b]$ can be subdivided into subintervals $a=a_0<a_1<a_2< \cdots <a_N=b$ such that either $f\ge 0$ over $[a_{i-1},a_i]$ or $f\le 0$ over $[a_{i-1},a_i]$. Set $A_i=\displaystyle\int_{a_{i-1}}^{a_i} f(t) dt$.

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

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

73. Suppose the average value of $f$ over $[a,b]$ is 1 and the average value of $f$ over $[b,c]$ is 1 where $a<c<b$. Show that the average value of $f$ over $[a,c]$ is also 1.

74. Suppose that $[a,b]$ can be partitioned. taking $a=a_0<a_1< \cdots < a_N=b$ such that the average value of $f$ over each subinterval $[a_{i-1},a_i]=1$ is equal to 1 for each $i=1\, \cdots , N$. Explain why the average value of $f$ over $[a,b]$ is also equal to 1.

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

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

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

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

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

80. Estimate $\displaystyle\int_0^1 $t$ dt$ 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 $\displaystyle\int_0^1 $t$ dt$?

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

82. From the graph of $\sin (2\pi x)$ shown:

1. Explain why $\displaystyle\int_0^1 \sin (2\pi t) dt=0$.
2. Explain why, in general, $\displaystyle\int_a^{a+1} \sin (2\pi t) dt=0$ for any value of $a$.

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

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