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6.5. IMPROPER INTEGRALS
(a) Determine the fraction of callers who wait between 5 and 10 minutes.
(b) Determine the fraction of callers who wait between 10 and 20 minutes.
(c) Next, let’s study how the fraction who wait up to a certain number of minutes:
i. What is the fraction of callers who wait between 0 and 5 minutes?
ii. What is the fraction of callers who wait between 0 and 10 minutes?
iii. Between 0 and 15 minutes? Between 0 and 20?
(d) Let F(b) represent the fraction of callers who wait between 0 and b minutes. Find
a formula for F(b) that involves a definite integral, and then use the First FTC to
find a formula for F(b) that does not involve a definite integral.
(e) What is the value of the limit lim
b→∞
F(b)? What is its meaning in the context of the
problem?
⊲⊳
Improper Integrals Involving Unbounded Intervals
In light of our example with light bulbs that fail, as well as with the problem involving
customer wait time in Preview Activity 6.5, we see that it is natural to consider questions
where we desire to integrate over an interval whose upper limit grows without bound. For
example, if we are interested in the fraction of light bulbs that fail within the first b months
of use, we know that the expression
b
0
0.3e
−0.3t dt
measures this value. To think about the fraction of light bulbs that fail eventually, we
understand that we wish to find
lim
b→∞
b
0
0.3e
−0.3t dt,
for which we will also use the notation
∞
0
0.3e
−0.3t dt.
(6.5)
Note particularly that we are studying the area of an unbounded region, as pictured in
Figure 6.20.
Anytime we are interested in an integral for which the interval of integration is
unbounded (that is, one for which at least one of the limits of integration involves ∞), we