Double Integrals

Double Integrals & Fubini

A double integral computes the signed volume under a surface $z = f(x,y)$ over a region $R$: $\iint_R f(x,y)\,dA$.

To evaluate, use iterated integrals by integrating one variable at a time:

$$\iint_R f\,dA = \int_a^b\int_{g_1(x)}^{g_2(x)} f\,dy\,dx$$

Fubini's theorem states that if $f$ is continuous on $R$, you can integrate in either order. Sketch $R$ first to choose the simpler order.

Common pitfall: The inner limits may depend on the outer variable, never the reverse. Writing $\int_0^y \dots$ as the outer integral is a classic setup error.

Applications & Center of Mass

Double integrals extend single-variable calculus to find average values and physical centers.

Worked insight: Always sketch the region $R$ to set correct limits. The key skill is setting up boundaries properly before integrating.

Key takeaway: If an integral looks impossible in one order, swap the limits using Fubini's theorem. Inner limits are always functions of the outer variable.