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Further Integration Techniques
and Applications of the Integral
Copyright © Cengage Learning. All rights reserved.
14.5 Improper Integrals and Applications
Copyright © Cengage Learning. All rights reserved.
Improper Integrals and Applications
All the definite integrals we have seen so far have had the
form f(x) dx, with a and b finite and f(x) piecewise
continuous on the closed interval [a, b].
If we relax one or both of these requirements somewhat,
we obtain what are called improper integrals. There are
various types of improper integrals.
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Integrals in Which a Limit of
Integration is Infinite
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Integrals in Which a Limit of Integration is Infinite
Integrals in which one or more limits of integration are
infinite can be written as
Let’s concentrate for a moment on the first form,
. What does the
mean here?
As it often does, it means that we are to take a limit as
something gets large. Specifically, it means the limit as the
upper bound of integration gets large.
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Integrals in Which a Limit of Integration is Infinite
Improper Integral with an Infinite Limit of Integration
We define
provided the limit exists. If the limit exists, we say that
converges.
Otherwise, we say that
diverges.
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Integrals in Which a Limit of Integration is Infinite
Similarly, we define
provided the limit exists.
Finally, we define
for some convenient a, provided both integrals on the right
converge.
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Integrals in Which a Limit of Integration is Infinite
Quick Example
Converges
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Example 1 – Future Sales of CDs
In 2006, music downloads were starting to make inroads
into the sales of CDs. Approximately 140 million CD albums
were sold in the first quarter of 2006, and sales declined by
about 3.5% per quarter for the following
2 years.
Suppose that this rate of decrease were to continue
indefinitely. How many CD albums, total, would be sold
from the first quarter of 2006 on?
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Example 1 – Solution
Recall that the total sales between two dates can be
computed as the definite integral of the rate of sales.
So, if we wanted the sales between the first quarter of 2006
and a time far in the future, we would compute
with a large M, where s(t) is the quarterly sales t quarters
after the first quarter of 2006.
Because we want to know the total number of CD albums
sold from the first quarter of 2006 on, we let M →
that is, we compute
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Example 1 – Solution
cont’d
Because sales of CD albums are decreasing by 3.5% per
quarter, we can model s(t) by
s(t) = 140(0.965)t million CD albums
where t is the number of quarters since the first quarter of
2006.
Total sales from the first
quarter of 2006 on
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Example 1 – Solution
cont’d
≈ 3929.6 million CD albums.
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Integrals in Which the Integrand
Becomes Infinite
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Example 2 – Integrand Infinite at One Endpoint
Calculate
Solution:
Notice that the integrand approaches
as x approaches
0 from the right and is not defined at 0. This makes the
integral an improper integral.
Figure 22 shows the region
whose area we are trying to
calculate; it extends infinitely
vertically rather than horizontally.
Figure 22
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Example 2 – Solution
cont’d
Now, if 0 < r < 1, the integral
is a proper
integral because we avoid the bad behavior at 0. This
integral gives the area shown in Figure 23.
If we let r approach 0 from
the right, the area in
Figure 23 will approach
the area in Figure 22.
Figure 23
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Example 2 – Solution
cont’d
So, we calculate
= 2.
Thus, we again have an infinitely long region with finite
area.
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Integrals in Which the Integrand Becomes Infinite
Improper Integral in Which the Integrand Becomes Infinite
If f(x) is defined for all x with a < x  b but approaches
as x approaches a, we define
provided the limit exists.
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Integrals in Which the Integrand Becomes Infinite
Similarly, if f(x) is defined for all x with a ≤ x < b but
approaches
as x approaches b, we define
provided the limit exists. In either case, if the limit exists,
we say that
converges. Otherwise, we say that
diverges.
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