Chapter 5 | Integration
599
Figure 5.38 The indicated area can be calculated by evaluating a definite integral using substitution.
2
5.34
Evaluate ∫
e 2 x dx .
0
Example 5.42 Growth of Bacteria in a Culture
Suppose the rate of growth of bacteria in a Petri dish is given by q ( t ) =3 t , where t is given in hours and q ( t ) is given in thousands of bacteria per hour. If a culture starts with 10,000 bacteria, find a function Q ( t ) that gives the number of bacteria in the Petri dish at any time t . How many bacteria are in the dish after 2 hours?
Solution We have
Q ( t ) = ⌠
t dt = 3 t ln3
⌡ 3
+ C .
Then, at t =0 we have Q (0) =10= 1 ln3
+ C , so C ≈9.090 and we get
Q ( t ) = 3 t ln3
+9.090.
At time t =2, we have
2 ln3
Q (2) = 3
+9.090
=17.282.
After 2 hours, there are 17,282 bacteria in the dish.
5.35 From Example 5.42 , suppose the bacteria grow at a rate of q ( t ) =2 t . Assume the culture still starts with 10,000 bacteria. Find Q ( t ). How many bacteria are in the dish after 3 hours?
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