Download 480 Chapter 8: Techniques of Integration 70. Find the volume of the

480
Chapter
8:
Integration
Techniques
of
70. Find the volume of the solid generated by revolving the region about the
x-axis.
Plot the integrand (sin t)/t for t > 0. Is the sine-integral function
everywhere increasing or decreasing? Do you think Si (x) = 0 for x > 0
? Check your answers by graphing the function Si (x) for 0 < x < 25.
71. Find the area of the region that lies between the curves y = secx and y —
tanx from x = 0 to x = IT/2.
Explore the convergence of
72. The region in Exercise 71 is revolved about the x-axis to generate a solid.
dt.
a. Find the volume of the solid.
b. Show that the inner and outer surfaces of the solid have infinite area.
73. Estimating the value of a convergent improper integral whose domain is
infinite
If it converges, what is its value?
Error function
The function
erf (x)
74
J0
a. Show that
' dx
-e^ < 0.000042,
and hence that f™ e ^ dx < 0.000042. Explain why this means that /0°°
e~%1 dx can be replaced by
e~x~ dx without introducing an error of
—— dt.
J o V77
If it converges, what appears to be its value? You will see how to
confirm your estimate in Section 14.4, Exercise 41.
74. The infinite paint can or Gabriel's horn As Example 3 shows, the integral
Jx (dx/x) diverges. This means that the integral
77. Normal probability distribution
~dx,
+
rb
1 +
x^dX >
2jr
The function f i x )
=
277
which measures the surface area of the solid of revolution traced out by
revolving the curve y = 1/x, 1 < x, about the x-axis, diverges also. By
comparing the two integrals, we see that, for every finite value b > 1,
xV
important applications in
b. Explore the convergence of
lUj b. Evaluate J0 e x dx numerically.
l7T
dt,
has
a. Plot the error function for 0 < x < 25.
magnitude greater than 0.000042.
277^1
2e~
V 77
called the error function,
probability and statistics.
is called the normal probability density function with mean /x and
standard deviation cr. The number JUL tells where the distribution is
centered, and a measures the "scatter" around the mean. From the theory
of probability, it is known that
j\ X* dX'
f
f ( x ) dx = 1.
1.
In what follows, let \x — 0 and a
Y\ a. Draw the graph of /. Find the intervals on which / is increasing, the
intervals on which / is decreasing, and any local extreme values and
where they occur.
b. Evaluate
/ /(x) dx
J-n
However, the integral
for n = 1,2, and 3.
771 — ) dx
c. Give a convincing argument that
for the volume of the solid converges.
(Hint: Show that 0 < f ( x ) < e~x/1 for x > 1, and for b > 1,
a. Calculate it.
b. This solid of revolution is sometimes described as a can that does not
hold enough paint to cover its own interior. Think about that for a
moment. It is common sense that a finite amount of paint cannot cover
an infinite surface. But if we fill the horn with paint (a finite amount),
then we will have covered an infinite surface. Explain the apparent
contradiction.
75. Sine-integral function
3
e^2dx->0
78. Show that if /(x) is integrable on every interval of real numbers and a and b
are real numbers with a < b , then
a.
The integral
Si(x) =
f ( x ) dx and
f-oo f i x )
sm t
dt,
called the sine-integral function, has important applications in optics.
/(x) dx = 1.
as b — >co.)
b.
dx and
f ( x ) dx both converge if and only if
/(x) dx both converge.
f ( x ) dx + fa°° f ( x ) dx =
the integrals involved converge.
f i x ) dx +
f i x ) dx when