Double Integrals

The central idea

For an integrable scalar function $f$ on a planar region $D$, $\iint_Df\,dA$ is the limit of sums $\sum f(x_i^,y_i^)\Delta A_i$. If $f\ge0$, it gives volume under $z=f$; with signs it gives net accumulation. Fubini's theorem permits iterated integration under standard continuity or integrability hypotheses.

Definitions, hypotheses, and notation

For continuous functions on closed bounded regions, ordinary Fubini conditions are satisfied. More singular functions may require absolute integrability before order can be freely exchanged. Type I regions use $g_1(x)\le y\le g_2(x)$; Type II regions use horizontal bounds. A region can be both, but one order may require fewer pieces.

Double integrals also compute mass, probability, and moments when $f$ is a density. Dividing a weighted integral by total mass produces a centroid coordinate, so units distinguish volume under a graph from a physically weighted planar total.

Conceptual meaning

A double integral weights every small area element by a local density. Integrating one variable first accumulates along a slice; the outer integral then combines slices across the region. Order is a description of how the same planar set is swept.

A dependable method and decision rule

1. Sketch and describe $D$ before choosing bounds.
2. Decide an order that represents each slice with simple limits.
3. Hold the outer variable constant during the inner integral.
4. Evaluate the resulting one-variable integral with correct outer bounds.
5. Use sign, area, and units to verify the total.

Fully worked example

Common mistakes and why they fail