Integrals for functions of single variables can be understand as the area under the curve. Similarly, double integral of a function of two variables
We can use Riemann sum to define double integral: suppose we have a region
As Iterated Integral
See also: Fubini’s theorem
We can also compute
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.