Question

An agricultural sprinkler distributes water in a circular pattern of radius 100 ft. It supplies water to a depth of e^-r feet per hour at a distance of r feet from the sprinkler. If 0 < R ≤ 100, what is the total amount of water supplied per hour to the region inside the circle of radius R centered at the sprinkler?

Answer 1

A circle of radius $R$ ft has area $\pi R^2$ sq ft. For any fixed $R$, water will reach a depth of $e^{-R}$ ft. You can think of the total volume of water supplied within a radius $R$ of the sprinkler as the volume of the "cylinder" with "height" given by $e^{-r}$ for some $0$.

This volume is

$\displaystyle\int_0^{2\pi}\int_0^Rre^{-r}\,\mathrm dr\,\mathrm d\theta=2\pi(1-e^{-R}(R+1))$

cu ft.