Write an integral expression for the volume of a sphere with radius 4. Use horizontal slices and let *h* be the depth of the slice below the top of the sphere.

Answer

We slice the sphere horizontally, with *h* the depth of the slice below the top of the sphere. Each slice has thickness Δ *h*.

If we cut the sphere in half along the *xz*-plane we can see half the sphere and half the slice. The radius of the slice is *x* where *x*^{2} + *z*^{2} = 4^{2}. Notice that* z* = 4 –* h*.

Since *z* = 4 – *h*, the Pythagorean Theorem says

*x*^{2} + (4 – *h*)^{2} = 4^{2}

so

This means the area of the circular side of the slice is

The volume of a slice is

π(4^{2}– (4 – *h*)^{2}) Δ *h*.

and, the volume of the sphere is

We can simplify this integral using the sphere's symmetry to

We could solve either integral, but it's up to you to solve the one that's easiest. We recommend the second one.