The relevant piece of mathematics describing when you can and cannot swap integrals is "Fubini's Theorem". You may learn the full details of Fubini's Theorem in an analysis course, …

Double integrals do more than find volume under three-dimensional graphs. Here we cover other uses, a more general notation for double integrals, and explain the "feel" of double integration.

What makes double integrals tricky is finding the bounds in non-rectangular regions. Here we go through what that means and practice a few examples.

Double integrals 6 Double integrals with variable bounds Finding bounds of regions Switching bounds on double integrals

A double integral is just one regular integral inside of another. Thus, you can use integration techniques on the inside integral, then use integration techniques for the outside integral.

There are many ways to extend the idea of integration to multiple dimensions: some examples include Line integrals, double integrals, triple integrals, and surface integrals.

In all of the double integrals we've done so far, the boundaries on x and y were fixed. Now we'll see what happens when the boundaries on x and y are variables.

Let's evaluate the double integrals with y=x^2 as one of the boundaries. Created by Sal Khan.

If you have a two-variable function described using polar coordinates, how do you compute its double integral?

In principle, the idea of a surface integral is the same as that of a double integral, except that instead of "adding up" points in a flat two-dimensional region, you are adding up points on a …