Double Integrals

A double integral is simply a way to add up something that's spread over a flat region, like rain falling on a field or charge sitting on a plate. The symbol $\iint_R f(x, y)\,dA$ looks intimidating, but it's just three small pieces: $R$ (the region you're covering), $dA$ (a tiny patch of that region) and $f(x, y)$ (how much stuff there is per unit area at that spot). Multiply each patch by its value of $f$, then add up all the patches. That's the whole idea; the rest of this unit is about doing that adding efficiently.

Take a look at how the single integral worked: chop $[a, b]$ into thin strips, multiply each width by a height, add, and let the strips get thinner. This is the same Riemann sum one dimension up. The strips become small rectangles in the plane, each strip's area becomes a box's volume, and the sum of the boxes closes in on the volume under the surface.

What the total means depends on what $f$ is:

• With $f \ge 0$ a height, $\iint_R f\,dA$ is the volume of the solid between the surface and the region $R$.
• With $f = 1$, every box has height $1$, so the total is just the area of $R$.
• Divided by the area of $R$, it's the average value of $f$ over $R$, exactly like $\frac{1}{b - a}\int_a^b f\,dx$ in one variable.

A common mistake. It's tempting to read every double integral as a volume. That's only true when $f \ge 0$. Where $f$ is negative the boxes count as negative, just as area below the axis did for single integrals, so $\iint_R f\,dA$ is really a net amount.

The midpoint rule estimates it by evaluating $f$ at the center of each small rectangle. It's surprisingly good: in the picture below, nine squares already land within $0.04$ of the exact volume, because over each square the surface rises on one side of the center about as much as it falls on the other. The next lesson shows how to get the exact value with two ordinary integrals.

Total rainfall over a county from a rainfall-rate map, total mass of a plate from its density, and the expected value of a function of two random variables are all double integrals.