45

Now we set up the following inequality
E
S
n
≤
24
·
1
5
180
n
4
≤
1
1000
Manipulating this inequality we have
n
4
≥
400
3
. Thus we write our final answer
as
Let
n
be the smallest
even
integer greater than
4
q
400
3
.
Using a calculator, we see that
n
= 4 subintervals is sufficient to guarantee that
our approximation is within
1
1000
of ln 2.
Note.
In our final answer, we had to write “Let
n
be the smallest even integer
.
.
.
”
Unlike the previous example, we had to specify that
n
was an even
integer because we were using Simpson’s Rule, which requires an even number
of subintervals.
8.2
Improper Integrals
There are two types of integrals that are described as
improper
.
I.
f
(
x
) is a bounded function, but the interval of integration is unbounded.
For instance, in the graph below the interval is [
a,
∞
).
II.
g
(
x
) is an unbounded function, but the interval of integration is bounded.
For instance, in the graph below the interval is [0
,
1], but the function has
a vertical asymptote at
x
= 0.
46

Question
: Can we define the area under the graph in either scenario?
I. For any
c > a
,
R
c
a
f
(
x
)
dx
is defined. If
R
c
a
f
(
x
)
dx
approaches a value as
c
increases without bound, then we define
Z
∞
a
f
(
x
)
dx
= lim
c
→∞
Z
c
a
f
(
x
)
dx
II. Suppose we wish to evaluate
R
b
a
g
(
x
)
dx
,
g
(
x
) is continuous on (
a, b
], and
g
(
x
) has a vertical asymptote at
x
=
a
. The
R
b
c
g
(
x
)
dx
is defined for all
a < c
≤
b
. If
R
a
c
g
(
x
)
dx
approaches a value as
c
approaches
a
from above,
then we define
Z
b
a
g
(
x
)
dx
=
lim
c
→
a
+
Z
b
c
g
(
x
)
dx
In either scenario, if the limit exists as a real number then we say the integral
converges
. Otherwise it
diverges
.
8.2.1
Type I Improper Integrals
We will begin by investigating integrals in which our function is bounded, but
we are integrating over an unbounded interval.
Example 8.3.
Determine whether
Z
∞
1
1
x
2
dx
converges or diverges. If it con-
verges, to what value does it converge?
We wish to find the area of the region shaded in the graph below. Is this an
infinite amount of area? Or is the amount of area under the curve bounded by
a value as we let our upper bound increase?
47