Two maths grads, a statistician and a mathematician, let's see if I can learn something here today!
Let's say I want to find the volume of a rectangular block (say 2x3x5, though a mathematician might say a x b x c), ok so we multiply base x height x length to get the volume. This can be viewed as taking the area of one surface (say 2 x 3), then summing each infinitesimally thin slice of this area along the length (integrating) to obtain the volume. This is the basic idea behind volume integration.
Lets now say I want to find the surface area of a sphere of radius R (4pi.r^2) by integration. Using spherical polar coordinates, I make a small change in two directions (e theta, e phi) to give an infinitesimally small section of the surface area (a very small square?) then by summing these sections in both directions I obtain the surface area for the sphere.
To find the volume for a sphere (4.pi.r^3), I first find the surface area as above, then sum along the e r direction from the origin to the radius.
That's what I want to do, but how do I go about it mathematically? (it's one of the things I've not understood despite having "done it " at least twice!).
