Multiple Integration

Multiple Integrals & Rectangles

A double integral sums a function over a 2D region, extending single-variable integration from intervals to areas:

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

Iterated integration evaluates this via nested single integrals. For a rectangular region $[a,b] \times [c,d]$:

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

Common pitfall: Using constant limits for non-rectangular regions. If the region is a triangle below $y = x$, inner limits must be functions ($0$ to $x$). Copying rectangle limits integrates the wrong shape.

Regions, Economics & Jacobian

For non-rectangular regions bounded by curves, the inner limits become functions:

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

Economic application: In multi-good settings, consumer surplus and producer surplus extend from single integrals to double integrals over regions where willingness to pay exceeds market price.

Change of variables: Coordinate transformations use the Jacobian determinant absolute value $|J|$:

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