Net change/word problem/integrals

Question

Water flows into a reservoir at a rate of 1500 + 7t2 gallons per hour. since there is a tiny hole in the reservoir, the water leaks out at a rate of 4t + t2 gallons per hour. What is the net change in the amount of water in the reservoir over the time interval [1,5]?

Answer 1

Let A(t) = amount of water in the reservoir at time t hours

 

Then A'(t) = rate of change in the amount of water (in gal/hour) at time t

 

                = (rate in) - (rate out)

 

                = (1500 + 7t²) - (4t + t²) = 6t² - 4t + 1500

 

(net change in amount of water from t=1 to t=5) = A(5) - A(1)

 

               = ∫_{from 1 to 5} A'(t)dt  (using the Fundamental Theorem of Calculus)

 

               = ∫_{from 1 to 5)} (6t² - 4t + 1500) dt

 

               = (2t³ - 2t² + 1500t)_{from 1 to 5}

 

               = 7700 - 1500 = 6200 gal