A circle of radius [tex]R[/tex] ft has area [tex]\pi R^2[/tex] sq ft. For any fixed [tex]R[/tex], water will reach a depth of [tex]e^{-R}[/tex] ft. You can think of the total volume of water supplied within a radius [tex]R[/tex] of the sprinkler as the volume of the "cylinder" with "height" given by [tex]e^{-r}[/tex] for some [tex]0<r\le R[/tex].
This volume is
[tex]\displaystyle\int_0^{2\pi}\int_0^Rre^{-r}\,\mathrm dr\,\mathrm d\theta=2\pi(1-e^{-R}(R+1))[/tex]
cu ft.
