6.5. IMPROPER INTEGRALS
377
6.5 Improper Integrals
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What are improper integrals and why are they important?
• What does it mean to say that an improper integral converges or diverges?
• What are some typical improper integrals that we can classify as convergent or
divergent?
Introduction
Another important application of the definite integral regards how the likelihood of certain
events can be measured. For example, consider a company that manufactures incandescent
light bulbs, and suppose that based on a large volume of test results, they have determined
that the fraction of light bulbs that fail between times t = a and t = b of use (where t is
measured in months) is given by
b
a
0.3e
−0.3t dt.
For example, the fraction of light bulbs that fail during their third month of use is given by
3
2
0.3e
−0.3t dt = −e
−0.3t
3
2
= −e
−0.9 + e
−0.6
≈ 0.1422.
Thus about 14.22% of all lightbulbs fail between t = 2 and t = 3. Clearly we could adjust
the limits of integration to measure the fraction of light bulbs that fail during any time
period of interest.
Preview Activity 6.5. A company with a large customer base has a call center that
receives thousands of calls a day. After studying the data that represents how long callers
wait for assistance, they find that the function p(t) = 0.25e −0.25t models the time customers
wait in the following way: the fraction of customers who wait between t = a and t = b
minutes is given by
b
a
p(t) dt.
Use this information to answer the following questions.
377
6.5 Improper Integrals
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What are improper integrals and why are they important?
• What does it mean to say that an improper integral converges or diverges?
• What are some typical improper integrals that we can classify as convergent or
divergent?
Introduction
Another important application of the definite integral regards how the likelihood of certain
events can be measured. For example, consider a company that manufactures incandescent
light bulbs, and suppose that based on a large volume of test results, they have determined
that the fraction of light bulbs that fail between times t = a and t = b of use (where t is
measured in months) is given by
b
a
0.3e
−0.3t dt.
For example, the fraction of light bulbs that fail during their third month of use is given by
3
2
0.3e
−0.3t dt = −e
−0.3t
3
2
= −e
−0.9 + e
−0.6
≈ 0.1422.
Thus about 14.22% of all lightbulbs fail between t = 2 and t = 3. Clearly we could adjust
the limits of integration to measure the fraction of light bulbs that fail during any time
period of interest.
Preview Activity 6.5. A company with a large customer base has a call center that
receives thousands of calls a day. After studying the data that represents how long callers
wait for assistance, they find that the function p(t) = 0.25e −0.25t models the time customers
wait in the following way: the fraction of customers who wait between t = a and t = b
minutes is given by
b
a
p(t) dt.
Use this information to answer the following questions.