Calculus AB rate integration

Question

2. A water treatment plant processes gray water at a rate given by r(t) = 16e ^ (0.2t) gallons per year, where is the measured in years for Write an expression that gives the amount of gray water processed by the treatment plant during the time interval 0 <= t <= 5

Answer 1

Let f(t) = amount of gray water treated in t years

Then, f'(t) = r(t) = 16e^0.2t.

The amount of water processed from t = 0 to t = 5 is f(5) - f(0).

By the Fundamental Theorem of Calculus, f(5) - f(0) = ∫_{from 0 to 5)} f'(t)dt =

16∫_{(from 0 to 5)} e^0.2tdt = 80e^02t_{(from 0 to 5)} = 80(e - 1)