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 since we're squaring that square root, the radical is going to disappear, leaving us with a very easy integral to compute. No calculators required. |