Double Integrals

A double integral computes the volume under $z = f(x,y)$ over $R$:

$$\iint_R f(x,y)\,dA$$

Iterated integrals — Evaluate 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 — If $f$ is continuous on $R$, integrate in either order.

Applications

• Area: $\iint_R 1\,dA$
• Average value: $\bar{f} = \frac{1}{\text{Area}(R)}\iint_R f\,dA$
• Mass: $M = \iint_R \rho\,dA$
• Centre of mass: $\bar{x} = \frac{1}{M}\iint_R x\rho\,dA$

Sketch $R$ and choose the integration order that gives simpler limits.

Key insight: The key skill is setting up the correct limits of integration.

Common pitfall: In iterated integrals the inner limits may depend on the outer variable, never the reverse. Writing $\int_0^{y}\ldots$ as the outer integral is the classic setup error — sketch the region first.

Geometric Meaning of the Double Integral

The double integral $\iint_R f(x,y)\,dA$ computes the signed volume between the surface $z = f(x,y)$ and the $xy$-plane over region $R$.

To evaluate, integrate iteratively:

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

The base region $R$ determines the limits. Choosing the right order can greatly simplify the calculation.

Double integrals generalize single-variable integrals to two variables — the key is correct limits.