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Chapter 5 | Integration
Example 5.43 Fruit Fly Population Growth
Suppose a population of fruit flies increases at a rate of g ( t ) =2 e 0.02 t , in flies per day. If the initial population of fruit flies is 100 flies, how many flies are in the population after 10 days? Solution Let G ( t ) represent the number of flies in the population at time t . Applying the net change theorem, we have
G (10) = G (0)+ ∫ 0 10
2 e 0.02 t dt
⎦ | 0 10
⎡ ⎣ 2
e 0.02 t ⎤
10
=100+
0.02
⎦ | 0
⎡ ⎣ 100 e 0.02 t ⎤
=100+
=100+100 e 0.2 −100 ≈122.
There are 122 flies in the population after 10 days.
5.36 Suppose the rate of growth of the fly population is given by g ( t ) = e 0.01 t , and the initial fly population is 100 flies. How many flies are in the population after 15 days?
Example 5.44 Evaluating a Definite Integral Using Substitution Evaluate the definite integral using substitution: ⌠ ⌡ 1 2 e 1/ x x 2 dx .
Solution This problem requires some rewriting to simplify applying the properties. First, rewrite the exponent on e as a power of x , then bring the x 2 in the denominator up to the numerator using a negative exponent. We have ⌠ ⌡ 1 2 e 1/ x x 2 dx = ∫ 1 2 e x −1 x −2 dx . Let u = x −1 , the exponent on e . Then du =− x −2 dx − du = x −2 dx . Bringing the negative sign outside the integral sign, the problem now reads
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