5128_Ch08_434-471

55. (b) V ln 2
A(x)dx ln 2
(p/4) e 2x dx
55. Each cross section of the solid infinite horn shown in the figure
cut by a plane perpendicular to the x-axis for x ln 2 is a
circular disc with one diameter reaching from the x-axis to the
curve y e x .
y
2
y = e x
1
0
ln 2
(a) Find the area of a typical cross section. A(x) (p/4) e 2x
(b) Express the volume of the horn as an improper integral.
(c) Find the volume of the horn. p/2
56. Normal Probability Distribution Function In Section 7.5,
we encountered the bell-shaped normal distribution curve that is
the graph of
1 ( ) 2
f x 1 x m
2 s
,
2
the normal probability density function with mean m and
standard deviation s. The number m tells where the distribution
is centered, and s measures the “scatter” around the mean.
From the theory of probability, it is known that
f x dx 1.
In what follows, let m 0 and s 1.
(a) Draw the graph of f. Find the intervals on which f is
increasing, the intervals on which f is decreasing, and any local
extreme values and where they occur.
(b) Evaluate
n
f x dx for n 1, 2, 3.
n
(c) Give a convincing argument that
f x dx 1.
(Hint: Show that 0 f x e x2 for x 1, and for b 1,
e x2 dx→0 as b→.)
b
57. Approximating the Value of 1 e x2 dx
(a) Show that
6
e x2 dx
e 6x dx 4 10 17 .
1 1
31. 0 1 e x e x on [1, ), converges because 1
1 e x dx converges
34.
1 1
32. 0 x 3
1 x 3 on [1, ), converges because 1
1 x 3 dx converges
33. 0 1 x 2 cos x
on [p, ), diverges because 1 dx diverges
x
p x
6
x
Section 8.4 Improper Integrals 469
(b) Writing to Learn Explain why
1
e x2 dx 6
e x2 dx
with error of at most 4 10 17 .
(c) Use the approximation in part (b) to estimate the value of
1 e x2 dx. Compare this estimate with the value displayed in
Figure 8.19.
(d) Writing to Learn Explain why
0
e x2 dx 3
e x2 dx
with error of at most 0.000042.
Extending the Ideas
58. Use properties of integrals to give a convincing argument that
Theorem 6 is true.
59. Consider the integral
f n 1
x n e x dx
0
where n 0.
(a) Show that 0
x n e x dx converges for n 0, 1, 2.
(b) Use integration by parts to show that f n 1 nfn.
(c) Give a convincing argument that 0
x n e x dx converges for
all integers n 0.
60. Let f x x
sin t
dt.
0
t
(a) Use graphs and tables to investigate the values of f x
as x→.
(b) Does the integral 0
sin xx dx converge? Give a
convincing argument.
61. (a) Show that we get the same value for the improper integral in
Example 5 if we express
1
dxx 1
2
1
dxx 2
1
0
and then evaluate these two integrals.
1
,
1
dxx 2
(b) Show that it doesn’t matter what we choose for c in
(Improper Integrals with Infinite Integration Limits, part 3)
f x dx c
f x dx
f x dx.
dx
x 4
2
1
dx
0
x 4
21
1
dx
0
x 4
2
1
dx
1
x and
4
1
1
0
x 4
1
1
x2 on [1, ), converges because 1
1 x 2 dx converges
c