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.

This is similar to any 2D example of integrals, where we typically draw rectangles with a height of y and a width of dx. For this 3D object, Sal drew a solid which had a height of x*y^2, width of y, …

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.

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

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.

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