31.1 Introduction

Surface and volume integrals, and similar integrals in higher dimensions are encountered in many subjects.

They can be used to determine areas of surfaces and volumes, quantities such as moments of inertia that are of interest in mechanics, moments of probability distributions, masses and charges in regions, potentials of a given charge distribution, flows through surfaces, and many other things as well.

We will consider two types of integrals here: flux and volume integrals. Area integrals in two dimensions can be considered as easy special cases of flux integrals.

Here are some examples of integrals you might encounter:

Find moment of inertia of a right circular cylinder of height h radius r and constant density .

Find volume of an ellipsoid with axis lengths A, B and C.

Find the electric potential a of a uniform charge distribution on the surface of a sphere.

Find the surface area of a cone like figure between 0 and h with radius r = z².

Surfaces can be described in several different ways.

You may have a qualitative description of the surface (it is a plane or the surface of a sphere or cylinder or cone, centered at some location etc.), or it can be defined by an equation: f(x, y, z) = 0.

Or you may have a parametric expression for each of x, y and z in terms of two parameters.

The easiest case occurs when your surface is described by an equation and you can solve the equation to obtain say z as an explicit function of x and y, (z = z(x, y)) so that you can use x and y as your two parameters.