5.2E: Exercises for Section 5.2

In exercises 1 - 4, express the limits as integrals.

1) \(\displaystyle \lim_{n→∞}\sum_{i=1}^n(x^∗_i)Δx\) over \([1,3]\)

2) \(\displaystyle \lim_{n→∞}\sum_{i=1}^n(5(x^∗_i)^2−3(x^∗_i)^3)Δx\) over \([0,2]\)

Answer

\(\displaystyle ∫^2_0(5x^2−3x^3)\,dx\)

3) \(\displaystyle \lim_{n→∞}\sum_{i=1}^n\sin^2(2πx^∗_i)Δx\) over \([0,1]\)

4) \(\displaystyle \lim_{n→∞}\sum_{i=1}^n\cos^2(2πx^∗_i)Δx\) over \([0,1]\)

Answer

\(\displaystyle ∫^1_0\cos^2(2πx)\,dx\)

In exercises 5 - 10, given \(L_n\) or \(R_n\) as indicated, express their limits as \(n→∞\) as definite integrals, identifying the correct intervals.

5) \(\displaystyle L_n=\frac{1}{n}\sum_{i=1}^n\frac{i−1}{n}\)

6) \(\displaystyle R_n=\frac{1}{n}\sum_{i=1}^n\frac{i}{n}\)

Answer

\(\displaystyle ∫^1_0x\,dx\)

7) \(\displaystyle Ln=\frac{2}{n}\sum_{i=1}^n(1+2\frac{i−1}{n})\)

8) \(\displaystyle R_n=\frac{3}{n}\sum_{i=1}^n(3+3\frac{i}{n})\)

Answer

\(\displaystyle ∫^6_3x\,dx\)

9) \(\displaystyle L_n=\frac{2π}{n}\sum_{i=1}^n2π\frac{i−1}{n}\cos(2π\frac{i−1}{n})\)

10 \(\displaystyle R_n=\frac{1}{n}\sum_{i=1}^n(1+\frac{i}{n})\log((1+\frac{i}{n})^2)\)

Answer

\(\displaystyle ∫^2_1x\log(x^2)\,dx\)

In 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)

Answer

\( 1+2⋅2+3⋅3=14\)

13)

14)

Answer

\(1−4+9=6\)

15)

16)

Answer

\(1−2π+9=10−2π\)

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

17) \(\displaystyle ∫^3_0(3−x)\,dx\)

18) \(\displaystyle ∫^3_2(3−x)\,dx\)

Answer

The integral is the area of the triangle, \(\frac{1}{2}.\)

19) \(\displaystyle ∫^3_{−3}(3−|x|)\,dx\)

20) \(\displaystyle ∫^6_0(3−|x−3|)\,dx\)

Answer

The integral is the area of the triangle, \(9.\)

21) \(\displaystyle ∫^2_{−2}\sqrt{4−x^2}\,dx\)

22) \(\displaystyle ∫^5_1\sqrt{4−(x−3)^2}\,dx\)

Answer

The integral is the area \(\frac{1}{2}πr^2=2π.\)

23) \(\displaystyle ∫^{12}_0\sqrt{36−(x−6)^2}\,dx\)

24) \(\displaystyle ∫^3_{−2}(3−|x|)\,dx\)

Answer

The integral is the area of the “big” triangle less the “missing” triangle, \(9−\frac{1}{2}.\)

In 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 \([0,8]\)

Answer

\( L=2+0+10+5+4=21,\; R=0+10+10+2+0=22,\; \dfrac{L+R}{2}=21.5\)

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 \( [−4,4]\)

Answer

\( L=0+4+0+4+2=10,\;R=4+0+2+4+0=10,\;\dfrac{L+R}{2}=10\)

Suppose that \(\displaystyle ∫^4_0f(x)\,dx=5\) and \(\displaystyle ∫^2_0f(x)\,dx=−3\), and \(\displaystyle ∫^4_0g(x)\,dx=−1\) and \(\displaystyle ∫^2_0g(x)\,dx=2\). In exercises 29 - 34, compute the integrals.

29) \(\displaystyle ∫^4_0(f(x)+g(x))\,dx\)

30) \(\displaystyle ∫^4_2(f(x)+g(x))\,dx\)

Answer

\(\displaystyle ∫^4_2f(x)\,dx+∫^4_2g(x)\,dx=8−3=5\)

31) \(\displaystyle ∫^2_0(f(x)−g(x))\,dx\)

32) \(\displaystyle ∫^4_2(f(x)−g(x))\,dx\)

Answer

\(\displaystyle ∫^4_2f(x)\,dx−∫^4_2g(x)\,dx=8+3=11\)

33) \(\displaystyle ∫^2_0(3f(x)−4g(x))\,dx\)

34) \(\displaystyle ∫^4_2(4f(x)−3g(x))\,dx\)

Answer

\(\displaystyle 4∫^4_2f(x)\,dx−3∫^4_2g(x)\,dx=32+9=41\)

In exercises 35 - 38, use the identity \(\displaystyle ∫^A_{−A}f(x)\,dx=∫^0_{−A}f(x)\,dx+∫^A_0f(x)\,dx\) to compute the integrals.

35) \(\displaystyle ∫^π_{−π}\frac{\sin t}{1+t^2}dt\) (Hint: \(\displaystyle \sin(−t)=−\sin(t))\)

36) \(\displaystyle ∫^{\sqrt{π}}_\sqrt{−π}\frac{t}{1+\cos t}dt\)

Answer

The integrand is odd; the integral is zero.

37) \(\displaystyle ∫^3_1(2−x)\,dx\) (Hint: Look at the graph of \(f\).)

38) \(\displaystyle ∫^4_2(x−3)^3\,dx\) (Hint: Look at the graph of \(f\).)

Answer

The integrand is antisymmetric with respect to \(x=3.\) The integral is zero.

In exercises 39 - 44, given that \(\displaystyle ∫^1_0x\,dx=\frac{1}{2},\;∫^1_0x^2\,dx=\frac{1}{3},\) and \(\displaystyle ∫^1_0x^3\,dx=\frac{1}{4}\), compute the integrals.

39) \(\displaystyle ∫^1_0(1+x+x^2+x^3)\,dx\)

40) \(\displaystyle ∫^1_0(1−x+x^2−x^3)\,dx\)

Answer

\(\displaystyle 1−\frac{1}{2}+\frac{1}{3}−\frac{1}{4}=\frac{7}{12}\)

41) \(\displaystyle ∫^1_0(1−x)^2\,dx\)

42) \(\displaystyle ∫^1_0(1−2x)^3\,dx\)

Answer

\(\displaystyle ∫^1_0(1−6x+12x^2−8x^3)\,dx=1−6\left( \frac{1}{2} \right)+12\left(\frac{1}{3}\right)−8\left(\frac{1}{4}\right)=1-3+4-2=0\)

43) \(\displaystyle ∫^1_0\left(6x−\tfrac{4}{3}x^2\right)\,dx\)

44) \(\displaystyle ∫^1_0(7−5x^3)\,dx\)

Answer

\(7−\frac{5}{4}=\frac{23}{4}\)

In exercises 45 - 50, use the comparison theorem.

45) Show that \(\displaystyle ∫^3_0(x^2−6x+9)\,dx≥0.\)

46) Show that \(\displaystyle ∫^3_{−2}(x−3)(x+2)\,dx≤0.\)

Answer

The integrand is negative over \([−2,3].\)

47) Show that \(\displaystyle ∫^1_0\sqrt{1+x^3}\,dx≤∫^1_0\sqrt{1+x^2}\,dx\).

48) Show that \(\displaystyle ∫^2_1\sqrt{1+x}\,dx≤∫^2_1\sqrt{1+x^2}\,dx.\)

Answer

\(x≤x^2\) over \([1,2]\), so \(\sqrt{1+x}≤\sqrt{1+x^2}\) over \([1,2].\)

49) Show that \(\displaystyle ∫^{π/2}_0\sin tdt≥\frac{π}{4}\) (Hint: \(\sin t≥\frac{2t}{π}\) over \( [0,\frac{π}{2}])\)

50) Show that \(\displaystyle ∫^{π/4}_{−π/4}\cos t\,dt≥π\sqrt{2}/4\).

Answer

\(\cos(t)≥\dfrac{\sqrt{2}}{2}\). Multiply by the length of the interval to get the inequality.

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

51) \( f(x)=x^2,\; a=−1,\; b=1\)

52) \( f(x)=x^5,\; a=−1,\; b=1\)

Answer

\(f_{ave}=0;\; c=0\)

53) \( f(x)=\sqrt{4−x^2},\; a=0,\; b=2\)

54) \(f(x)=3−|x|,\; a=−3,\; b=3\)

Answer

\(\frac{3}{2}\) when \(c=±\frac{3}{2}\)

55) \(f(x)=\sin x,\; a=0,\; b=2π\)

56) \( f(x)=\cos x,\; a=0,\; b=2π\)

Answer

\(f_{ave}=0;\; c=\dfrac{π}{2},\; \dfrac{3π}{2}\)

In exercises 57 - 60, 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 \( 2(\sqrt{e}−1).\)

Answer

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

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

60) [T] \(y=\dfrac{x+1}{\sqrt{4−x^2}}\) over the interval \([−1,1]\); the exact solution is \(\dfrac{π}{6}\).

Answer

\(L_{100}×(\dfrac{1}{2})=0.5178,\; R_{100}×(\dfrac{1}{2})=0.5294\)

In exercises 61 - 64, 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 \(\frac{1}{4}(e^4−1).\)

Answer

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

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

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

Answer

\(L_1×(\frac{1}{π})=1.352,L_{10}×(\frac{1}{π})=−0.1837,L_{100}×(1π)=−0.2956.\) The exact answer \(≈−0.303,\) so \(L_{100}\) is not accurate to the first decimal.

65) Suppose that \(\displaystyle A=∫^{2π}_0\sin^2t\,dt\) and \(\displaystyle B=∫^{2π}_0\cos^2t\,dt.\) Show that \(A+B=2π\) and \(A=B.\)

66) Suppose that \(\displaystyle A=∫^{π/4}_{−π/4}\sec^2 t\,dt=π\) and \(\displaystyle B=∫^{π/4}_{−π/}4\tan^2 t\,dt.\) Show that \(A−B=\dfrac{π}{2}\).

Answer

Use \(\tan^2 θ+1=\sec^2 θ.\) Then, \(\displaystyle B−A=∫^{π/4}_{−π/4}1\,dx=\frac{π}{2}.\)

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

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

Answer

\(\displaystyle ∫^{2π}_0\cos^2t\,dt=π,\) so divide by the length \(2π\) of the interval. \(\cos^2t\) has period \(π\), so yes, it is true.

69) Explain why the graphs of a quadratic function (parabola) \(p(x)\) and a linear function \(ℓ(x)\) can intersect in at most two points. Suppose that \(p(a)=ℓ(a)\) and \(p(b)=ℓ(b)\), and that \(\displaystyle ∫^b_ap(t)\,dt>∫^b_aℓ(t)dt\). Explain why \(\displaystyle ∫^d_cp(t)>∫^d_cℓ(t)\,dt\) whenever \( a≤c<d≤b.\)

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

Answer

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 \(a=a_0<a_1<a_2<⋯<a_N=b\) such that either \(f≥0\) over \([a_{i−1},a_i]\) or \(f≤0\) over \([a_{i−1},a_i]\). Set \(\displaystyle A_i=∫^{a_i}_{a_{i−1}}f(t)\,dt.\)

a. Explain why \(\displaystyle ∫^b_af(t)\,dt=A_1+A_2+⋯+A_N.\)

b. Then, explain why \(\displaystyle ∫^b_af(t)\,dt≤∫^b_a|f(t)|\,dt.\)

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

Answer

If \(f(t_0)>g(t_0)\) for some \(t_0∈[a,b]\), 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 \(\displaystyle ∫^d_df(t)\,dt>∫^d_cg(t)\,dt\)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 \(a<c<b\). Show that the average value of \(f\) over \([a,c]\) is also \(1.\)

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

Answer

The integral of f over an interval is the same as the integral of the average of f over that interval. Thus, \(\displaystyle ∫^b_af(t)\,dt=∫^{a_1}_{a_0}f(t)\,dt+∫^{a_2}_a{1_f}(t)\,dt+⋯+∫^{a_N}_{a_{N+1}}f(t)\,dt=∫^{a_1}_{a_0}1\,dt+∫^{a_2}_{a_1}1\,dt+⋯+∫^{a_N}_{a_{N+1}}1\,dt\)
\( =(a_1−a_0)+(a_2−a_1)+⋯+(a_N−a_{N−1})=a_N−a_0=b−a\).
Dividing through by \(b−a\) gives the desired identity.

75) Suppose that for each \(i\) such that \( 1≤i≤N\) one has \(\displaystyle ∫^i_{i−1}f(t)\,dt=i\). Show that \(\displaystyle ∫^N_0f(t)\,dt=\frac{N(N+1)}{2}.\)

76) Suppose that for each \(i\) such that \(1≤i≤N\) one has \(\displaystyle ∫^i_{i−1}f(t)\,dt=i^2\). Show that \(\displaystyle ∫^N_0f(t)\,dt=\frac{N(N+1)(2N+1)}{6}\).

Answer

\(\displaystyle ∫^N_0f(t)\,dt=\sum_{i=1}^N∫^i_{i−1}f(t)\,dt=\sum_{i=1}^Ni^2=\frac{N(N+1)(2N+1)}{6}\)

77) [T] Compute the left and right Riemann sums \(\displaystyle L_{10}\) and \(R_{10}\) and their average \(\dfrac{L_{10}+R_{10}}{2}\) for \( f(t)=t^2\)over \( [0,1]\). Given that \(\displaystyle ∫^1_0t^2\,dt=1/3\), to how many decimal places is \( \dfrac{L_{10}+R_{10}}{2}\) accurate?

78) [T] Compute the left and right Riemann sums, \(L_10\) and \(R_{10}\), and their average \(\dfrac{L_{10}+R_{10}}{2}\) for \( f(t)=(4−t^2)\) over \([1,2]\). Given that \(\displaystyle ∫^2_1(4−t^2)\,dt=1.66\), to how many decimal places is \(\dfrac{L_{10}+R_{10}}{2}\) accurate?

Answer

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

79) If \(\displaystyle ∫^5_1\sqrt{1+t^4}\,dt=41.7133...,\) what is \(\displaystyle ∫^5_1\sqrt{1+u^4}\,du?\)

80) Estimate \(\displaystyle ∫^1_0t\,dt\) 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 \(\displaystyle ∫^1_0t\,dt?\)

Answer

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

81) Estimate \(\displaystyle ∫^1_0t\,dt\) by comparison with the area of a single rectangle with height equal to the value of \(t\) at the midpoint \(t=\dfrac{1}{2}\). How does this midpoint estimate compare with the actual value \(\displaystyle ∫^1_0t\,dt?\)

82) From the graph of \(\sin(2πx)\) shown:

a. Explain why \(\displaystyle ∫^1_0\sin(2πt)\,dt=0.\)

b. Explain why, in general, \(\displaystyle ∫^{a+1}_a\sin(2πt)\,dt=0\) for any value of \(a\).

Answer

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 \(\displaystyle ∫^1_0f(t)\,dt=0?\)

84) If f is 1-periodic and \(\displaystyle ∫10f(t)\,dt=A,\) is it necessarily true that \(\displaystyle ∫^{1+a}_af(t)\,dt=A\) for all \(A\)?

Answer

Yes, the integral over any interval of length 1 is the same.