380
6.5. IMPROPER INTEGRALS
in an expression that depends on b).
iii. Now, use your work from (ii.) to evaluate the limit given by
lim
b→∞
b
1
1
x
dx.
(b) Next, we investigate
∞
1
1
x 3/2 dx.
i. Use the First FTC to determine the exact values of
10
1
1
x 3/2 dx,
1000
1
1
x 3/2 dx,
and
100000
1
1
x 3/2 dx. Then, use your calculator to compute a decimal
approximation of each result.
ii. Use the First FTC to evaluate the definite integral
b
1
1
x 3/2 dx (which
results in an expression that depends on b).
iii. Now, use your work from (ii.) to evaluate the limit given by
lim
b→∞
b
1
1
x 3/2 dx.
(c) Plot the functions y =
1
x and y =
1
x 3/2 on the same coordinate axes for the
values x = 0 . . . 10. How would you compare their behavior as x increases
without bound? What is similar? What is different?
(d) How would you characterize the value of
∞
1
1
x dx? of
∞
1
1
x 3/2 dx? What does
this tell us about the respective areas bounded by these two curves for x ≥ 1?
⊳
Convergence and Divergence
Our work so far has suggested that when we consider a nonnegative function f on an
interval [1, ∞], such as f (x) =
1
x or f (x) =
1
x 3/2 , there are at least two possibilities for the
value of lim b→∞
b
1
f (x) dx: the limit is finite or infinite. With these possibilities in mind,
we introduce the following terminology.
If f (x) is nonnegative for x ≥ a, then we say that the improper integral
∞
a
f (x) dx
converges provided that
lim
b→∞
b
a
f (x) dx
exists and is finite. Otherwise, we say that
∞
a
f (x) dx diverges.
We normally restrict our interest to improper integrals for which the integrand is
6.5. IMPROPER INTEGRALS
in an expression that depends on b).
iii. Now, use your work from (ii.) to evaluate the limit given by
lim
b→∞
b
1
1
x
dx.
(b) Next, we investigate
∞
1
1
x 3/2 dx.
i. Use the First FTC to determine the exact values of
10
1
1
x 3/2 dx,
1000
1
1
x 3/2 dx,
and
100000
1
1
x 3/2 dx. Then, use your calculator to compute a decimal
approximation of each result.
ii. Use the First FTC to evaluate the definite integral
b
1
1
x 3/2 dx (which
results in an expression that depends on b).
iii. Now, use your work from (ii.) to evaluate the limit given by
lim
b→∞
b
1
1
x 3/2 dx.
(c) Plot the functions y =
1
x and y =
1
x 3/2 on the same coordinate axes for the
values x = 0 . . . 10. How would you compare their behavior as x increases
without bound? What is similar? What is different?
(d) How would you characterize the value of
∞
1
1
x dx? of
∞
1
1
x 3/2 dx? What does
this tell us about the respective areas bounded by these two curves for x ≥ 1?
⊳
Convergence and Divergence
Our work so far has suggested that when we consider a nonnegative function f on an
interval [1, ∞], such as f (x) =
1
x or f (x) =
1
x 3/2 , there are at least two possibilities for the
value of lim b→∞
b
1
f (x) dx: the limit is finite or infinite. With these possibilities in mind,
we introduce the following terminology.
If f (x) is nonnegative for x ≥ a, then we say that the improper integral
∞
a
f (x) dx
converges provided that
lim
b→∞
b
a
f (x) dx
exists and is finite. Otherwise, we say that
∞
a
f (x) dx diverges.
We normally restrict our interest to improper integrals for which the integrand is