Let's slice the sphere vertically. Since each slice is a circle, we center the sphere at the origin and use *x* as the variable of integration. The slices have thickness Δ *x*. If we cut the sphere in half down the middle at the *xz*-plane, we can see what half of the sphere and half of a slice look like. The radius of the slice is z. We cut the sphere along the *xz*-plane, where y = 0, so *x*^{2} + z^{2} = 5^{2}. This means The volume of the slice is then The variable *x* goes from -5 to 5, so the volume of the sphere is Alternately, since the sphere is symmetric we could find half its volume and then multiply by 2. If we do things this way we get the expression Notice that we would have to do a more complicated integral involving a square root. If we had to solve this integral right now, karma and your brother would be getting the last laugh. Since we don't, we just managed to set up a nice integral to look at and have a calculator solve. |