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Unformatted text preview: 10.7 IMPROPER INTEGRALS
1 629 and
0 dx = lim x4/5 c→0+ 1 c dx = lim 5x1/5 x4/5 c→0+ 1 c = lim [5 − 5c1/5 ] = 5.
c→0+ The improper integral converges and
1 −2 dx = 5 + 5(21/5 ) ∼ 10. 74. = x 4 /5 EXERCISES 10.7
Evaluate the improper integrals that converge.
∞ 1.
1 ∞ dx . x2 dx . 4 + x2 e dx,
px ∞ 2.
0 ∞ dx . 1 + x2 e
−px c In Exercises 35–36, use a graphing utility to draw the graph of the integrand in each of the improper integrals. Then use a CAS to determine whether the integral converges or diverges.
∞ 3.
0 ∞ 4.
0 1 dx, p > 0. 35. (a)
0 ∞ x dx. (16 + x2 )2 x dx. 16 + x4 x3 2−x x 2−x dx. dx. ∞ (b)
0 ∞ x2 dx. (16 + x2 )2 x dx. 16 + x2 1 2−x 1 2x − x 2 dx. dx. 5.
0 8 p > 0. 6.
0 1 dx √. x dx . x2 √ dx 1−x dx a2 − x 2 . . (c)
0 2 (d)
0 2 7.
0 1 dx . x2/3 √ dx 1 − x2 x 4 − x2 . dx. 36. (a)
0 2 8.
0 1 √ 3 √ (b)
0 2 √ (c)
0 (d)
0 √ 9.
0 2 10.
0 a 37. Evaluate
1 0 11.
0 √
∞ 12.
0 √
∞ sin−1 x dx 13.
e 1 ln x dx. x 14.
e ∞ dx . x ln x dx . x(ln x)2 dx . x2 − 1 15.
0 x ln x dx.
∞ 16.
e ∞ by using integration by parts even though the technique leads to an improper integral. 38. (a) For what values of r is
∞ 0 17.
−∞ ∞ dx . 1 + x2 dx . x2 dx . x(x + 1) x x2 − 9 dx. 18.
2 3 xr e−x dx 19.
−∞ ∞ 20. 22. dx . √ 3 3x − 1 1/ 3
0 convergent? (b) Use mathematical induction to show that
∞ 0 21.
1 5 x ex dx.
−∞ 4 xn e−x dx = n!, n = 1, 2, 3, . . . . 23.
3 3 √ 24.
1 dx . 2−4 x x dx. (1 + x2 )2 1 dx. ex + e−x 39. The integral
∞ 0 √ 25.
−3 1 dx . x(x + 1) x2 dx . −4 ∞ 1 dx x (1 + x) 26.
1 ∞ 27.
−3 ∞ 28.
−∞ 4 is improper in two distinct ways: the interval of integration is unbounded and the integrand is unbounded. If we rewrite the integral as
1 29.
0 ∞ cosh x dx. e
0 1 −x 30.
1 dx . 2 − 5x + 6 x cos x dx.
2 √
0 1 dx + x (1 + x) ∞ 1 √ 1 dx, x(1 + x) ∞ 31. 33.
0 sin x dx. 32.
0 ex √ dx. x √ π /2 34.
0 cos x dx. √ sin x then we have two improper integrals, the ﬁrst having an unbounded integrand and the second deﬁned on an unbounded interval. If each of these integrals converges with values L1 and L2 , respectively, then the original integral converges and has the value L1 + L2 . Evaluate the original integral. 630 CHAPTER 10 SEQUENCES; INDETERMINATE FORMS; IMPROPER INTEGRALS 40. Evaluate
∞ 1 57. (a) Show that the improper integral 1 dx √ x x2 − 1
∞ 0 2x dx 1 + x2 using the method given in Exercise 39. 41. Show that if the region below the graph of f (x) = 1/x, x ≥ 1, is revolved about the xaxis, then the surface area of the resulting solid is inﬁnite (see Example 3). 42. Sketch the graphs of y = sec x and y = tan x for 0 ≤ x < π/2. Calculate the area of the region between the two curves. 43. Let be the region bounded by the coordinate axes, the √ graph of y = 1/ x, and the line x = 1. (a) Sketch . (b) Show that has ﬁnite area and ﬁnd it. (c) Show that if is revolved about the xaxis, the solid obtained does not have ﬁnite volume. 44. Let be the region between the graph of y = 1/(1 + x2 ) and the xaxis, x ≥ 0. (a) Sketch . (b) Find the area of . (c) Find the volume of the solid obtained by revolving about the xaxis. (d) Find the volume of the solid obtained by revolving about the yaxis. 45. Let be the region bounded by the curve y = e−x and the xaxis, x ≥ 0. (a) Sketch . (b) Find the area of . (c) Find the volume of the solid obtained by revolving about the xaxis. (d) Find the volume obtained by revolving about the yaxis. (e) Find the lateral surface area of the solid in part (c). 46. What point would you call the centroid of the region in Exercise 45? Does Pappus’s theorem work in this instance? 47. Let be the region bounded by the curve y = e−x and the xaxis, x ≥ 0. (a) Show that has ﬁnite area. (The area is √ actually 1 π , as you will see in Chapter 16.) (b) Calculate 2 the volume generated by revolving about the yaxis. 48. Let be the region bounded below by y(x2 + 1) = x, above by xy = 1, and to the left by x = 1. (a) Find the area of . (b) Show that the solid generated by revolving about the xaxis has ﬁnite volume. (c) Calculate the volume generated by revolving about the yaxis. 49. Let be the region bounded by the curve y = x−1/4 and the xaxis, 0 < x ≤ 1. (a) Sketch . (b) Find the area of . (c) Find the volume of the solid obtained by revolving about the xaxis. (d) Find the volume of the solid obtained by revolving about the yaxis. 50. Prove the validity of the comparison test (10.7.2). In Exercises 51–56, use the comparison test (10.7.2) to determine whether the integral converges.
∞
2 diverges. Thus, the improper integral 2x dx 1 + x2 −∞ diverges. (b) Show that lim 58. Show that
∞ b→∞ ∞ 2x dx = 0. 2 −b 1 + x and b (a)
−∞ sin x dx diverges
b (b) lim b→∞ −b sin x dx = 0. 59. Calculate the arc distance from the origin to the point (x(θ1 ), y(θ1 )) along the exponential spiral r = a ecθ . (Take a > 0, c > 0.) 60. The function 1 f (x ) = √ 2π
x −∞ e−t 2 /2 dt is important in statistics. Prove that the integral on the right converges for all real x. Exercises 59–62: Laplace transforms. Let f be continuous on [0, ∞). The Laplace transform of f is the function F deﬁned by F (s) =
0 ∞ e−sx f (x) dx. 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 . 61. f (x) = 1. 63. f (x) = cos 2x. 62. f (x) = x. 64. f (x) = eax . Exercises 63–66: Probability density functions. A nonnegative function f deﬁned on ( − ∞, ∞) is called a probability density function if
∞ −∞ f (x) dx = 1. 65. Show that the function f deﬁned by f (x ) = 6x/(1 + 3x2 )2 0 x≥0 x<0 51.
1 ∞ dx. √ 1 + x5 (1 + x ) ln x dx. x2
5 −1/6 x ∞ 52.
1 ∞ 2 −x2 dx. 53.
0 ∞ dx. 54.
π ∞ sin2 2x dx. x2 √ dx x + 1 ln x . is a probability density function. 66. Let k > 0. Show that the function f (x ) = ke−kx x ≥ 0 0 x < 0, 55.
1 56.
e CHAPTER HIGHLIGHTS 631 is a probability density function. It is called the exponential density function. 67. The mean of a probability density function f is deﬁned as the number µ=
∞ where µ is the mean. Calculate the standard deviation for the exponential density function. 69. (Useful later) Let f be a continuous, positive, decreasing function on [1, ∞). Show that
∞ x f (x) dx.
−∞ 1 f (x) dx converges iff the sequence
n Calculate the mean for the exponential density function. 68. The standard deviation of a probability density function f is deﬁned as the number σ=
∞ −∞ 1/ 2 an =
1 f (x) dx (x − µ) f (x) dx
2 converges. CHAPTER HIGHLIGHTS
10.1 The Least Upper Bound Axiom for each α > 0, lim upper bound, bounded above, least upper bound (p. 585) least upper bound axiom (p. 586) lower bound, bounded below, greatest lower bound (p. 587)
10.2 Sequences of Real Numbers 1 =0 nα n x for each real x, lim =0 n→∞ n! ln n lim =0 lim n1/n = 1 n→∞ n n→∞ xn for each real x, lim 1 − = ex n→∞ n Cauchy sequence (p. 610)
n→∞ sequence (p. 590) bounded above, bounded below, bounded (p. 591) increasing, nondecreasing, decreasing, nonincreasing (p. 591) recurrence relation (p. 594) It is sometimes possible to obtain useful information about a sequence yn = f (n) by applying the techniques of calculus to the function y = f (x).
10.3 Limit of a Sequence 10.5 The Indeterminate Form (0/0) L Hopital’s rule (0/0) (p. 611) ’ˆ Cauchy meanvalue theorem (p. 613)
10.6 The Indeterminate Form (∞/∞) ; other Indeterminate Forms limit of a sequence (p. 595) uniqueness of the limit (p. 597) convergent, divergent (p. 597) pinching theorem (p. 600) Every convergent sequence is bounded (p. 597); thus, every unbounded sequence is divergent. A bounded, monotonic sequence converges. (p. 598) Suppose that cn → c as n → ∞, and all the cn are in the domain of f . If f is continuous at c, then f (cn ) → f (c). (p. 601)
10.4 Some Important Limits L Hopital’s rule (∞/∞) (p. 616) ’ˆ ln x xk lim α = 0 lim x = 0 lim xx = 1 x→∞ x x→∞ e x →0 + other indeterminate forms: 0 · ∞, ∞ − ∞, 00 , 1∞ , ∞ ∞ (p. 617)
10.7 Improper Integrals integrals over inﬁnite intervals (p. 622) convergent, divergent (p. 623)
∞ 1 dx converges for p > 1 and diverges for p ≤ 1. xp for x > 0, n→∞ lim x1/n = 1; for x < 1 n→∞ lim xn = 0 a comparison test (p. 624) integrals of unbounded functions (p. 626) convergent, divergent (p. 627) ...
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 Spring '10
 SMITH
 Improper Integrals, Integrals

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