Chapter 6 - Applications of the Integral - 6.2 Setting Up Integrals: Volume, Density, Average Value - Exercises - Page 296: 5

$V$ = $\pi(Rh^{2}-\frac{1}{3}h^{3})$

let r = radius at height y $r$ = $\sqrt {R^{2}-(R-y)^{2}}$ $V$ = $\pi\int_0^{h}r^{2}dy$ $V$ = $\pi\int_0^{h}\sqrt {R^{2}-(R-y)^{2}}dy$ $V$ = $\pi\int_0^{h}(R^{2}-R^{2}+2Ry-y^{2})dy$ $V$ = $\pi\int_0^{h}(2Ry-y^{2})dy$ $V$ = $\pi(Ry^{2}-\frac{1}{3}y^{3})|_0^h$ $V$ = $\pi(Rh^{2}-\frac{1}{3}h^{3})$