# Integration using polar coordinates

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Integration using polar coordinates is a technique for solving integrals using polar coordinates. Sometimes an integral that is complicated in one set or coordinates, such as Cartesian coordinates become very easy or even trivial in polar coordinates.

## Example

Consider the integral



This can be converted into polar coordinated by multiplying  by itself, so that



This can be expressed as the double integral,



Now the usefulness of polar coordinates becomes apparent as in polar coordinates, . The bounds mean the integral is over the entire x-y plane, so  varies from  to  and  from  to . To convert the differentials, we must multiply by the Jacobian,  to get



This can be integrated by separating the integral. The  integral gives  and the  integral gives . This means the initial integral,  is .

## Other methods of integration

The primary methods of integration include: