Integral Calculus, The Volume of the Spheroid

The Volume of the Spheroid

In the last section we derived the volume and surface area of the unit sphere in n dimensions. Furthermore, a sphere of radius r has its volume scaled by rⁿ, while its surface area is scaled by r^n-1. But what happens if the sphere is scaled by different factors along its coordinates? For example, a circle could be stretched along the x axis and squashed along the y axis, giving an ellipse.

In general, a sphere that is scaled differently along its axes is a spheroid. To find the volume of such a shape, use integration by substitution to pull the integral back to the one that computes the volume of the unit sphere. This brings in a factor of r_i, where r_i is the "radius" along the i^th coordinate. For instance, the spheroid with equation x²+y²+16z² = 1 extends outward from the origin to x = ±1, y = ±1 and z = ±¼. Take the original volume of the sphere in 3 dimensions and multiply by 1×1×¼, giving a new volume of π/3.

A sphere that is squashed, like a flattened disk (see the above example), is called an oblate spheroid. A sphere that is stretched along one dimension, like a football with rounded ends, is a prolate spheroid, or an ellipsoid.

There is no convenient formula for the surface area of these spheroids in n dimensions. Formulas exist for oblate and prolate spheroids in 3 dimensions.