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.
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 x2 + z2 = 42. Notice that z = 4 – h.
Since z = 4 – h, the Pythagorean Theorem says
x2 + (4 – h)2 = 42
This means the area of the circular side of the slice is
The volume of a slice is
π(42– (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.