Multiple Integration

Integrating Over Regions

A double integral sums a function over a two-dimensional region — the natural extension of single-variable integration from intervals to areas.

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

Iterated integration evaluates a double integral as two nested single integrals. For a rectangular region $[a,b] \times [c,d]$:

$$\iint_R f(x,y)\, dA = \int_a^b \left(\int_c^d f(x,y)\, dy\right) dx$$

Fubini's theorem: if $f$ is continuous on the rectangle, the order of integration doesn't matter — integrate first with respect to $y$ then $x$, or vice versa. The result is the same.

Non-rectangular regions: if the region is bounded by curves, the limits of the inner integral become functions. For example, the region below $y = g(x)$ and above $y = h(x)$:

$$\int_a^b \int_{h(x)}^{g(x)} f(x,y)\, dy\, dx$$

Economic application — consumer surplus: the area between the demand curve $P = D(Q)$ and the market price $P^*$ is consumer surplus. In a two-good setting, the joint surplus involves a double integral over the region where willingness to pay exceeds the price.

$$CS = \int_0^{Q^*} \bigl(D(Q) - P^*\bigr)\, dQ$$

This extends to producer surplus, total welfare, and deadweight loss — all defined as areas (single integrals) or volumes (double integrals) between economic curves.

Change of variables: just as $u$-substitution works in one dimension, the Jacobian handles coordinate transformations in two dimensions:

$$\iint f(x,y)\, dx\, dy = \iint f\bigl(x(u,v), y(u,v)\bigr) |J|\, du\, dv$$

where $|J| = |\partial(x,y)/\partial(u,v)|$ is the absolute value of the Jacobian determinant. This is essential when the region is naturally circular (use polar coordinates) or when a linear substitution simplifies the integrand.

Tip: Fubini lets you choose the integration order — and the right choice can turn an impossible inner integral into a routine one. If the first order you try gets ugly, swap before you fight it.

Common pitfall: Keeping constant limits when the region isn't a rectangle. Over the triangle below $y = x$, the inner limits must be functions ($y$ from $0$ to $x$) — copying rectangle limits silently integrates over the wrong region.