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5.2E: Exercises for Section 5.2

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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 \([0,2]\)

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 \([0,1]\)

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

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

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

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

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

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.

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

17) \(\displaystyle ∫^3_0(3−x)\,dx\)

18) \(\displaystyle ∫^3_2(3−x)\,dx\)

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

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

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

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

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

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

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 \([0,8]\)

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 \(\displaystyle ∫^4_0f(x)\,dx=5\) 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) \(\displaystyle ∫^4_2(f(x)+g(x))\,dx\)

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

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

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

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

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

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 \(f\).)

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

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

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

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

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

In exercises 45 - 50, use the comparison theorem.

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

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

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

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

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

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) \( f(x)=x^5,\; a=−1,\; b=1\)

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

54) \(f(x)=3−|x|,\; a=−3,\; b=3\)

55) \(f(x)=\sin x,\; a=0,\; b=2π\)

56) \( f(x)=\cos x,\; a=0,\; b=2π\)

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 \( 2(\sqrt{e}−1).\)

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 \(\dfrac{π}{6}\).

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 \(\frac{1}{4}(e^4−1).\)

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 \( \dfrac{\cos(π^2)−1}{2π.}\)

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 \(A−B=\dfrac{π}{2}\).

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 \(1/2.\)

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?

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 \(x.\)

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 \(1.\)

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 \(\displaystyle ∫^N_0f(t)\,dt=\frac{N(N+1)(2N+1)}{6}\).

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?

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 \(\displaystyle ∫^1_0t\,dt?\)

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 \(a\).

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 \(A\)?

Home > INT2 > Chapter 5 > Lesson 5.2.4

Lesson 5.1.1, lesson 5.1.2, lesson 5.1.3, lesson 5.1.4, lesson 5.1.5, lesson 5.2.1, lesson 5.2.2, lesson 5.2.3, lesson 5.2.4, lesson 5.2.5, lesson 5.2.6.

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