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**HallsofIvy** Aren't you even going to try this yourself? Set up a coordinate system so that the center of the sphere is at (0, 0, 0). The equation of the sphere is $\displaystyle x^2+ y^2+ z^2= r^2$. Take the "cap" to be at the "top" (along the z-axis) so that the top is at (0, 0, r) and the base of the cap is at r- h. Slice that cap with planes parallel to the xy-plane. Each slice is a disk with center at (0, 0, z) and equation $\displaystyle x^2+ y^2= r^2- z^2$ so it has radius $\displaystyle \sqrt{r^2- z^2}$ and area $\displaystyle \pi(r^2- z^2)$. Taking each disk to have thickness "dz" The volume of each disk is $\displaystyle \pi(r^2- z^2)dz$ and the total area is $\displaystyle \pi\int_{r- h}^r r^2- z^2 dz$.