Double integrals are a way to integrate over a two-dimensional area. Among other things, they lets us compute the volume under a surface.

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

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.

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.

You hopefully have a little intuition now on what a double integral is or how we go about figuring out the volume under a surface. So let's actually compute it and I think it'll all become a lot …

We do need to replace dxdy (or dydx) by r dr d theta when converting from a double integral in rectangular coordinates to a double integral in polar coordinates.

Evaluate the given double integral between bound 1 and bound 2, which are functions, not numbers.

Triple integrals are the analog of double integrals for three dimensions. They are a tool for adding up infinitely many infinitesimal quantities associated with points in a three-dimensional region.