Theory

Introduction

The polar coordinate system uses parameters $(r,\theta )$ instead of $(x,y)$ to identify points in the plane. These coordinates are useful in problems having rotational symmetry. The two coordinate systems are related by these equations:

$$\begin{array}{rlrl}x& =r\cos (\theta )& r^2& =x^2+y^2\\ y& =r\sin (\theta )& \tan (\theta )& =y/x\end{array}$$

Area under polar graphs

There is a formula to compute the area of the “sector sum” formed by a polar graph.

$$A={\int }_a^b\frac{1}{2}r(\theta {)}^{2}d\theta$$

The area of a sector is $\frac{1}{2}{r}^2\theta$, so the area of an infinitesimal sector is $\frac{1}{2}{r}^{2}d\theta$, and the formula just sums these.