A definite integral accumulates contributions along an interval. A multiple integral extends the same idea to regions with two or more spatial dimensions.

For a scalar function $f(x,y)$ over a region $D$ in the plane, the double integral is written

$$\iint_D f(x,y),dA.$$

For a scalar function in three dimensions,

$$\iiint_V f(x,y,z),dV$$

accumulates over a volume $V$.

From small rectangles to a region

Divide a planar region into many small pieces of area $\Delta A_i$. A sum

$$\sum_i f(x_i^,y_i^)\Delta A_i$$

approximates the total accumulation. As the pieces become finer, the limiting value defines the double integral when the limit exists.

The same construction in three dimensions uses small volume elements.

Iterated integrals

Many multiple integrals can be evaluated one variable at a time. Over a rectangular region,

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

\int_a^b\int_c^d f(x,y),dy,dx.$$

The inner integral treats $x$ as a parameter; its result is then integrated with respect to $x$.

Bounds describe geometry

For nonrectangular regions, the integration bounds depend on the other variable. The order of integration determines how the region must be described.

Multiple integration therefore combines the accumulation idea of ordinary integration with a geometric description of the region being accumulated over.