Integration using polar coordinates

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

$I=\int _{-\infty }^{\infty }e^{-x^{2}}\,{\text{d}}x$

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

$I^{2}=\int _{-\infty }^{\infty }e^{-x^{2}}\,{\text{d}}x\int _{-\infty }^{\infty }e^{-y^{2}}\,{\text{d}}y$

This can be expressed as the double integral,

$I^{2}=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-(x^{2}+y^{2})}\,{\text{d}}x\,{\text{d}}x$

Now the usefulness of polar coordinates becomes apparent as in polar coordinates, $r^{2}=x^{2}+y^{2}$. The bounds mean the integral is over the entire x-y plane, so $r$ varies from ${\text{0}}$ to $\infty$ and $\phi$ from ${\text{0}}$ to $2\pi$. To convert the differentials, we must multiply by the Jacobian, $r$ to get

$I^{2}=\int _{r=0}^{r=\infty }\int _{\phi =0}^{\phi =2\pi }re^{-r^{2}}\,{\text{d}}r\,{\text{d}}\phi$

This can be integrated by separating the integral. The $\phi$ integral gives $2\pi$ and the $r$ integral gives ${\text{1}}$. This means the initial integral, $I$ is ${\sqrt {2\pi }}$.

Other methods of integration

The primary methods of integration include:

• Integration by parts
• Integration using polar coordinates
• Multiple integration
• Residue calculus (for definite integrals)
• Method of simultaneous convolutions
• Mellin transforms
• Inflation-restriction sequences