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Mathematics LibreTexts

8.7E: Exercises for Section 8.7

  • Gilbert Strang & Edwin “Jed” Herman
  • OpenStax

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In exercises 1 - 8, evaluate the following integrals. If the integral is not convergent, answer “It diverges.”

1) 42dx(x3)2

Answer
It diverges.

2) 014+x2dx

3) 2014x2dx

Answer
Converges to π2

4) 11xlnxdx

5) 1xexdx

Answer
Converges to 2e

6) xx2+1dx

Answer
It diverges.

7) Without integrating, determine whether the integral 11x3+1dx converges or diverges by comparing the function f(x)=1x3+1 with g(x)=1x3.

Answer
It converges since 1x3+1<1x3, and 11x3dx converges.

8) Without integrating, determine whether the integral 11x+1dx converges or diverges.

In exercises 9 - 25, determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.

9) 0excosxdx

Answer
Converges to 12.

10) 1lnxxdx

11) 10lnxxdx

Answer
Converges to 4.

12) 10lnxdx

13) 1x2+1dx

Answer
Converges to π.

14) 51dxx1

15) 22dx(1+x)2

Answer
It diverges.

16) 0exdx

Answer
Converges to 1.

17) 0sinxdx

Answer
It diverges.

18) ex1+e2xdx

19) 10dx3x

Answer
Converges to 1.5.

20) 20dxx3

21) 21dxx3

Answer
It diverges.

22) 10dx1x2

23) 301x1dx

Answer
It diverges.

24) 15x3dx

Answer
Converges to 2.5.

25) 535(x4)2dx

Answer
It diverges.

In exercises 26 and 27, determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.

26) 1dxx2+4x; compare with 1dxx2.

Answer
Both integrals converge since 1dxx2+4x1dxx2 which converges.  If an integral is smaller than another convergent integral, it must converge.

27) 1dxx1; compare with 1dxx.

Answer
Both integrals diverge since 1dxx11dxx which diverges.  If an integral is bigger than another divergent integral, it must diverge.

In exercises 28 - 38, evaluate the integrals. If the integral diverges, answer “It diverges.”

28) 1dxxe

Answer
Converges to 1e1.

29) 10dxxπ

Answer
It diverges.

30) 10dx1x

31) 10dx1x

Answer
It diverges.

32) 0dxx2+1

33) 11dx1x2

Answer
Converges to π.

34) 10lnxxdx

35) e0ln(x)dx

Answer
Converges to 0.

36) 0xexdx

37) x(x2+1)2dx

Answer
Converges to 0.

38) 0exdx

In exercises 39 - 44, evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.

39) 90dx9x

Answer
Converges to 6.

40) 127dxx2/3

41) 30dx9x2

Answer
Converges to π2.

42) 246dttt236

43) 40xln(4x)dx

Answer
Converges to 8ln(16)4.

44) 30x9x2dx

45) Evaluate t.5dx1x2. (Be careful!) (Express your answer using three decimal places.)

Answer
Converges to about 1.047.

46) Evaluate 41dxx21. (Express the answer in exact form.)

47) Evaluate 2dx(x21)3/2.

Answer
Converges to 1+23.

48) Find the area of the region in the first quadrant between the curve y=e6x and the x-axis.

49) Find the area of the region bounded by the curve y=7x2, the x-axis, and on the left by x=1.

Answer
A=7.0 units.2

50) Find the area under the curve y=1(x+1)3/2, bounded on the left by x=3.

51) Find the area under y=51+x2 in the first quadrant.

Answer
A=5π2 units.2

52) Find the volume of the solid generated by revolving about the x-axis the region under the curve y=3x from x=1 to x=.

53) Find the volume of the solid generated by revolving about the y-axis the region under the curve y=6e2x in the first quadrant.

Answer
V=3πunits3

54) Find the volume of the solid generated by revolving about the x-axis the area under the curve y=3ex in the first quadrant.

The Laplace transform of a continuous function over the interval [0,) is defined by F(s)=0esxf(x)dx (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of each of the following functions and give the domain of F.

55) f(x)=1

Answer
1s,s>0

56) f(x)=x

57) f(x)=cos(2x)

Answer
ss2+4,s>0

58) f(x)=eax

59) Use the formula for arc length to show that the circumference of the circle x2+y2=1 is 2π.

Answer
Answers will vary.

A function is a probability density function if it satisfies the following definition: f(t)dt=1. The probability that a random variable x lies between a and b is given by P(axb)=baf(t)dt.

60) Show that f(x)={0,ifx<07e7x,ifx0 is a probability density function.

61) Find the probability that x is between 0 and 0.3. (Use the function defined in the preceding problem.) Use four-place decimal accuracy.

Answer
0.8775

This page titled 8.7E: Exercises for Section 8.7 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.

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