Integrals for functions of single variables can be understand as the area under the curve. Similarly, double integral of a function of two variables can be understand as the volume under a surface.

We can use Riemann sum to define double integral: suppose we have a region in the xy-plane and a two-variable function . The double integral is defined as:

As Iterated Integral

See also: Fubini’s theorem

We can also compute using the method of slices: Let be the area of a slice parallel to the yz-plane at a given x-coordinate. Then the volume can be expressed as: . For a given , .

Thus, we can write the double integral as an iterated integral:

Note: for a non-rectangular region, the range of integration for may depend on , as the boundaries of the region may change according to

In certain cases, the region of integration can be more naturally described in the polar coordinate, so changing variables to polar coordinate and then setting up an iterated integral is more convenient to solve the problems.