Evaluating double integrals in polar coordinates is useful when the region of integration is more naturally described in polar form, such as circles, sectors, or regions with radial symmetry.

This method transforms the integral from Cartesian coordinates to polar coordinates .

Transformation

The relationship between Cartesian and polar coordinates is given by:

where and .

The area element in polar coordinates is . The additional factor of arrises from the Jacobian of the transformation from Cartesian to polar coordinates.

Double Integral Formula

To evaluate over a region in polar coordinates:

1. Express in terms of and .
2. Determine the limits for and that describe .
3. Use the formula:

Example

Evaluate where is the disk of radius 1 centred at the origin.

In polar coordinates, , and is , .