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 $[0,2]$

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$\displaystyle\int_0^2 (5x^2-3x^3) dx$

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 $[0,1]$

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$\displaystyle\int_0^1 \cos^2 (2\pi x) dx$

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

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\int_{0}^{1} x d x

$\displaystyle\int_0^1 x dx$

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

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

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$\displaystyle\int_3^6 x dx$

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

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\int_{1}^{2} x log (x^{2}) d x

$\displaystyle\int_1^2 x\log(x^2) dx$

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.

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.

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$1+2 \cdot 2+3 \cdot 3=14$

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].

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$1-4+9=6$

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.

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$1-2\pi +9=10-2\pi$

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

17. $\displaystyle\int_0^3 (3-x) dx$

18. $\displaystyle\int_2^3 (3-x) dx$

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The integral is the area of the triangle, $\frac{1}{2}$

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

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

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

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

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The integral is the area

\frac{1}{2} π r^{2} = 2 π

$\frac{1}{2}\pi r^2=2\pi$ .

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

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

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The integral is the area of the “big” triangle minus the “missing” triangle, $9-\frac{1}{2}$ .

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 $[0,8]$

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$L=2+0+10+5+4=21,R=0+10+10+2+0=22, \, \frac{L+R}{2}=21.5$

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 $[-4,4]$

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$L=0+4+0+4+2=10,R=4+0+2+4+0=10, \, \frac{L+R}{2}=10$

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

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$\displaystyle\int_2^4 f(x) dx + \displaystyle\int_2^4 g(x) dx=8-3=5$

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

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

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$\displaystyle\int_2^4 f(x) dx - \displaystyle\int_2^4 g(x) dx=8+3=11$

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

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

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$4 \displaystyle\int_2^4 f(x) dx - 3 \displaystyle\int_2^4 g(x) dx = 32+9=41$

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

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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 $f$ .)

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The integrand is antisymmetric with respect to $x=3$ . The integral is zero.

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

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$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}=\frac{7}{12}$

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

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

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$\displaystyle\int_0^1 (1-6x+12x^2-8x^3) dx = (x-3x^{2}+4x^{3}-2x^{4})=(1-3+4-2)=0$

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

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

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$7-\frac{5}{4}=\frac{23}{4}$

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

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The integrand is negative over $[-2,3]$ .

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

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$x \le x^2$ over $[1,2]$ , so $\sqrt{1+x} \le \sqrt{1+x^2}$ over $[1,2]$ .

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

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$\cos (t) \ge \frac{\sqrt{2}}{2}$ . Multiply by the length of the interval to get the inequality.

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. $f(x)=x^5, \, a=-1, \, b=1$

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$f_{\text{ave}}=0; \, c=0$

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

54. $f(x)=(3-|x|), \, a=-3, \, b=3$

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$\frac{3}{2}$ when $c= \pm \frac{3}{2}$

55. $f(x)= \sin x, \, a=0, \, b=2\pi$

56. $f(x)= \cos x, \, a=0, \, b=2\pi$

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$f_{\text{ave}}=0; \, c=\frac{\pi}{2},\frac{3\pi}{2}$

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 $2(\sqrt{e}-1)$ .

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$L_{100}=1.294, \, R_{100}=1.301$ ; the exact average is between these values.

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 $\frac{\pi }{6}$ .

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$L_{100} \times (\frac{1}{2})=0.5178, \, R_{100} \times (\frac{1}{2})=0.5294$

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 $\frac{1}{4}(e^4-1)$ .

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$L_1=0, \, L_{10} \times (\frac{1}{2})=8.743493, \, L_{100} \times (\frac{1}{2})=12.861728$ . The exact answer $\approx 26.799$ , so $L_{100}$ is not accurate.

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 $\dfrac{\cos (\pi^2)-1}{2\pi}$ .

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$L_1 \times (\frac{1}{\pi})=1.352, \, L_{10} \times (\frac{1}{\pi})=-0.1837, \, L_{100} \times (\frac{1}{\pi})=-0.2956$ . The exact answer $\approx -0.303$ , so $L_{100}$ is not accurate to first decimal.

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 $B-A=\frac{\pi }{2}$ .

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Use $\tan^2 \theta +1= \sec^2 \theta$ . Then, $B-A=\displaystyle\int_{−\pi/4}^{\pi/4} 1 dx = \frac{\pi}{2}$ .

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 $1/2$ .

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$\displaystyle\int_0^{2\pi} \cos^2 t dt = \pi$ , so divide by the length $2\pi$ of the interval. $\cos^2 t$ has period $\pi$ , so yes, it is true.

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?

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The integral is maximized when one uses the largest interval on which $p$ is nonnegative. Thus, $A=\frac{−b-\sqrt{b^2-4ac}}{2a}$ and $B=\frac{−b+\sqrt{b^2-4ac}}{2a}$ .

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

Explain why $\displaystyle\int_a^b f(t) dt = A_1+A_2+ \cdots +A_N$ .
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 $x$ .

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If $f(t_0)>g(t_0)$ for some $t_0 \in [a,b]$ , then since $f-g$ is continuous, there is an interval containing $t_0$ such that $f(t)>g(t)$ over the interval $[c,d]$ , and then $\displaystyle\int_c^d f(t) dt>\displaystyle\int_c^d g(t) dt$ over this interval.

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

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

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The integral of $f$ over an interval is the same as the integral of the average of $f$ over that interval. Thus, $\begin{array}{l} \displaystyle\int_a^b f(t) dt=\displaystyle\int_{a_0}^{a_1} f(t) dt + \displaystyle\int_{a_1}^{a_2} f(t) dt + \cdots + \displaystyle\int_{a_{N-1}}^{a_N} f(t) dt = \displaystyle\int_{a_0}^{a_1} 1 dt + \displaystyle\int_{a_1}^{a_2} 1 dt + \cdots + \displaystyle\int_{a_{N-1}}^{a_N} 1 dt \\ =(a_1-a_0)+(a_2-a_1)+ \cdots +(a_N-a_{N-1})=a_N-a_0=b-a. \end{array}$

Dividing through by $b-a$ gives the desired identity.

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 $\displaystyle\int_0^N f(t) dt=\frac{N(N+1)(2N+1)}{6}$ .

Show Solution

$\displaystyle\int_0^N f(t) dt=\underset{i=1}{\overset{n}{\Sigma}} \displaystyle\int_{i-1}^i f(t) dt=\underset{i=1}{\overset{n}{\Sigma}} i^2=\frac{N(N+1)(2N+1)}{6}$

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?

Show Solution

$L_{10}=1.815, \, R_{10}=1.515, \, \frac{L_{10}+R_{10}}{2}=1.665$ , so the estimate is accurate to two decimal places.

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 $\displaystyle\int_0^1 t dt$ ?

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The average is $1/2$ , which is equal to the integral in this case.

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:

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

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a. The graph is antisymmetric with respect to $t=\frac{1}{2}$ over $[0,1]$ , so the average value is zero. b. For any value of $a$ , the graph between $[a,a+1]$ is a shift of the graph over $[0,1]$ , so the net areas above and below the axis do not change and the average remains zero.

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 $A$ ?

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Integration: Supplemental Content

Problem Set: The Definite Integral

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