Answer to Question #101815 in Calculus for Seth

Question

Volume flow rate of water flowing in the pipe Q = 𝑑𝑉/𝑑𝑡 [m^3/ h] varies with time as follows: Q (t) = 5 + 2cos^2 (πt/12). How many cubic meters of water is flowing through the pipe through the day? (the unit of time is an hour)

Answer 1

Since $\frac{dV}{dt}=Q$ , we can calculate V as

$V=\int\limits_0^{24}Q(t)dt=\int\limits_0^{24}(5+2cos^2\frac{\pi t}{12}dt)=\left(5t+t+\frac{6sin\frac{\pi t}{6}}{\pi}\right)_0^{24}=144$