Double Integrals

_R f(x, y) dA = _a^b _c^d f dy dx — iterated integration, Fubini, general regions.

The idea

The double integral extends integration from an interval to a region of the plane.

Take a region $R$, cut it into small patches of area $\Delta A$, multiply each patch's area by the value of $f$ at a point of the patch, and add. As the patches shrink, the sums converge to the double integral: $\iint_R f(x, y)\, dA = \lim_{\Delta A \to 0} \sum f(x_i, y_i)\, \Delta A,$ which we read as the double integral over $R$ of $f$ with respect to area. When $f$ is positive, each term is the volume of a thin column standing on its patch, so the total is the volume between the graph $z = f(x, y)$ and the region $R$ beneath it. Where $f$ is negative the columns hang below and count against the total, so the value is a signed volume. Taking $f = 1$ gives every column height one, and the integral returns the plain area of $R$.

Exactly as in one variable, the definition is an honest description and a poor way to compute. In practice we integrate one variable at a time, reducing the double integral to two ordinary integrals nested one inside the other; the order of $dx$ and $dy$ in the notation says which variable comes first.

Ways to work on it

• Walkthrough. Evaluate a double integral one variable at a time, via iterated integrals and Fubini's theorem.
• Practice. Area of a triangle as a double integral over a non-rectangular region.
• Hardest. Evaluate a double integral over a rectangle with letters for limits.

Not sure where to start? Take the ten-question placement test.