Double Integrals

Volume under a surface

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $\iint_R f\,dA$ is the volume under $f$: the limit of sums of thin columns.
• Compute it as an iterated integral, one variable at a time; the order doesn’t matter (Fubini).
• A double integral in polar coordinates gives $\int e^{-x^2}dx = \sqrt\pi$.

Steps

1. A flat roof
2. Adding columns
3. Grid of columns
4. One variable at a time
5. Iterate
6. The Gaussian integral