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5.2E: Exercises for Section 5.2

( \newcommand{\kernel}{\mathrm{null}\,}\)

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

1) limnni=1(xi)Δx over [1,3]

2) limnni=1(5(xi)23(xi)3)Δx over [0,2]

Answer
20(5x23x3)dx

3) limnni=1sin2(2πxi)Δx over [0,1]

4) limnni=1cos2(2πxi)Δx over [0,1]

Answer
10cos2(2πx)dx

In exercises 5 - 10, given Ln or Rn as indicated, express their limits as n as definite integrals, identifying the correct intervals.

5) Ln=1nni=1i1n

6) Rn=1nni=1in

Answer
10xdx

7) Ln=2nni=1(1+2i1n)

8) Rn=3nni=1(3+3in)

Answer
63xdx

9) Ln=2πnni=12πi1ncos(2πi1n)

10 Rn=1nni=1(1+in)log((1+in)2)

Answer
21xlog(x2)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)

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.

Answer
1+22+33=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].

Answer
14+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)

A graph with three shaded parts. The first is a triangle with endpoints at (0, 0), (2, 0), and (1, 1), which corresponds to the function 1 - |x-1| over [0, 2] in quadrant 1. 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 a triangle with endpoints at (6, 0), (12, 0), and (9, 3), which corresponds to the function 3 - |x-9| over [6, 12].

Answer
12π+9=102π

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

17) 30(3x)dx

18) 32(3x)dx

Answer
The integral is the area of the triangle, 12.

19) 33(3|x|)dx

20) 60(3|x3|)dx

Answer
The integral is the area of the triangle, 9.

21) 224x2dx

22) 514(x3)2dx

Answer
The integral is the area 12πr2=2π.

23) 12036(x6)2dx

24) 32(3|x|)dx

Answer
The integral is the area of the “big” triangle less the “missing” triangle, 912.

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,L+R2=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,L+R2=10

Suppose that 40f(x)dx=5 and 20f(x)dx=3, and 40g(x)dx=1 and 20g(x)dx=2. In exercises 29 - 34, compute the integrals.

29) 40(f(x)+g(x))dx

30) 42(f(x)+g(x))dx

Answer
42f(x)dx+42g(x)dx=83=5

31) 20(f(x)g(x))dx

32) 42(f(x)g(x))dx

Answer
42f(x)dx42g(x)dx=8+3=11

33) 20(3f(x)4g(x))dx

34) 42(4f(x)3g(x))dx

Answer
442f(x)dx342g(x)dx=32+9=41

In exercises 35 - 38, use the identity AAf(x)dx=0Af(x)dx+A0f(x)dx to compute the integrals.

35) ππsint1+t2dt (Hint: sin(t)=sin(t))

36) ππt1+costdt

Answer
The integrand is odd; the integral is zero.

37) 31(2x)dx (Hint: Look at the graph of f.)

38) 42(x3)3dx (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 10xdx=12,10x2dx=13, and 10x3dx=14, compute the integrals.

39) 10(1+x+x2+x3)dx

40) 10(1x+x2x3)dx

Answer
112+1314=712

41) 10(1x)2dx

42) 10(12x)3dx

Answer
10(16x+12x28x3)dx=16(12)+12(13)8(14)=13+42=0

43) 10(6x43x2)dx

44) 10(75x3)dx

Answer
754=234

In exercises 45 - 50, use the comparison theorem.

45) Show that 30(x26x+9)dx0.

46) Show that 32(x3)(x+2)dx0.

Answer
The integrand is negative over [2,3].

47) Show that 101+x3dx101+x2dx.

48) Show that 211+xdx211+x2dx.

Answer
xx2 over [1,2], so 1+x1+x2 over [1,2].

49) Show that π/20sintdtπ4 (Hint: sint2tπ over [0,π2])

50) Show that π/4π/4costdtπ2/4.

Answer
cos(t)22. Multiply by the length of the interval to get the inequality.

In exercises 51 - 56, find the average value fave of f between a and b, and find a point c, where f(c)=fave

51) f(x)=x2,a=1,b=1

52) f(x)=x5,a=1,b=1

Answer
fave=0;c=0

53) f(x)=4x2,a=0,b=2

54) f(x)=3|x|,a=3,b=3

Answer
32 when c=±32

55) f(x)=sinx,a=0,b=2π

56) f(x)=cosx,a=0,b=2π

Answer
fave=0;c=π2,3π2

In exercises 57 - 60, approximate the average value using Riemann sums L100 and R100. How does your answer compare with the exact given answer?

57) [T] y=ln(x) over the interval [1,4]; the exact solution is ln(256)31.

58) [T] y=ex/2 over the interval [0,1]; the exact solution is 2(e1).

Answer
L100=1.294,R100=1.301; the exact average is between these values.

59) [T] y=tanx over the interval [0,π4]; the exact solution is 2ln(2)π.

60) [T] y=x+14x2 over the interval [1,1]; the exact solution is π6.

Answer
L100×(12)=0.5178,R100×(12)=0.5294

In exercises 61 - 64, compute the average value using the left Riemann sums LN for N=1,10,100. How does the accuracy compare with the given exact value?

61) [T] y=x24 over the interval [0,2]; the exact solution is 83.

62) [T] y=xex2 over the interval [0,2]; the exact solution is 14(e41).

Answer
L1=0,L10×(12)=8.743493,L100×(12)=12.861728. The exact answer 26.799, so L100 is not accurate.

63) [T] y=(12)x over the interval [0,4]; the exact solution is 1564ln(2).

64) [T] y=xsin(x2) over the interval [π,0]; the exact solution is cos(π2)12π.

Answer
L1×(1π)=1.352,L10×(1π)=0.1837,L100×(1π)=0.2956. The exact answer 0.303, so L100 is not accurate to the first decimal.

65) Suppose that A=2π0sin2tdt and B=2π0cos2tdt. Show that A+B=2π and A=B.

66) Suppose that A=π/4π/4sec2tdt=π and B=π/4π/4tan2tdt. Show that AB=π2.

Answer
Use tan2θ+1=sec2θ. Then, BA=π/4π/41dx=π2.

67) Show that the average value of sin2t over [0,2π] is equal to 1/2. Without further calculation, determine whether the average value of sin2t over [0,π] is also equal to 1/2.

68) Show that the average value of cos2t over [0,2π] is equal to 1/2. Without further calculation, determine whether the average value of cos2(t) over [0,π] is also equal to 1/2.

Answer
2π0cos2tdt=π, so divide by the length 2π of the interval. cos2t has period π, so yes, it is true.

69) Explain why the graphs of a quadratic function (parabola) p(x) and a linear function (x) can intersect in at most two points. Suppose that p(a)=(a) and p(b)=(b), and that bap(t)dt>ba(t)dt. Explain why dcp(t)dt>dc(t)dt whenever ac<db.

70) Suppose that parabola p(x)=ax2+bx+c opens downward (a<0) and has a vertex of y=b2a>0. For which interval [A,B] is BA(ax2+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=bb24ac2a and B=b+b24ac2a.

71) Suppose [a,b] can be subdivided into subintervals a=a0<a1<a2<<aN=b such that either f0 over [ai1,ai] or f0 over [ai1,ai]. Set Ai=aiai1f(t)dt.

a. Explain why baf(t)dt=A1+A2++AN.

b. Then, explain why baf(t)dtba|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)\,dtover 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^2over [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 graph of the function f(x) = sin(2pi*x) over [0, 2]. The function is shaded over [.7, 1] above the curve and below to x axis, over [1,1.5] under the curve and above the x axis, and over [1.5, 1.7] above the curve and under the x axis. The graph is antisymmetric with respect o t = ½ over [0,1].

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.

5.2E: Exercises for Section 5.2 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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