This post is second in a series of 3 beginning with “Coordinates in 3-space”. We have been taught that the volume of a sphere . We will go through two derivations of this, beginning with the Calculus version. We will make use of a theorem in calculus.
Suppose we have a sphere of radius . Then from the previous post involving spherical coordinates, we see that every point in the sphere is made of where , , and . These are exactly our limits of integration hence we just have to compute the following integral!