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

Graphical or geometric meaning

Vertical strips in the base region carry columns of height $x+2y$. The inner integral compresses each strip to one accumulated value; the outer integral adds those strip totals.

Common mistakes and why they fail

Verification and reasonableness checks

• Integrate in the opposite order on a rectangle.
• Bound the result between area times minimum and maximum of a continuous integrand.
• Set $f=1$ and recover the region's area.

Describe the region before choosing an order

A double integral accumulates density over area. For a Type I region, vertical slices give $a\le x\le b$ and $g_1(x)\le y\le g_2(x)$; a Type II description reverses the roles. Sketch the region, mark intersections, and decide whether one order avoids splitting or difficult antiderivatives. Under standard continuity or integrability hypotheses on a bounded region, Fubini's theorem justifies iterated evaluation. The inner limits may depend on the outer variable, but the outer limits must be constants in an elementary setup. Check area by setting the integrand equal to one. A negative result for a nonnegative integrand signals reversed bounds or an incorrect region description.

Practice

1. Evaluate $\int_0^1\int_0^1(x+y)dydx$.
2. What does $\iint_D1dA$ equal?
3. Can order always be swapped without changing bounds?

Connections and next steps

• Iterated Integrals
• Double Integrals in Polar Coordinates
• Triple Integrals