Area under a polar function — integration in polar coordinates

The familiar xy-plane uses Cartesian coordinates to represent points in a 2D space. Every point in space can be represented by a pair of (x, y) coordinates, representing the distance of the point from the origin in the x and y directions.

There is an alternative coordinate system, polar coordinates. In this system, the position of each point is represented by r (the straight line distance of the point from the origin) and θ (the angle the point makes with the x-axis at the origin). That is shown here:

It is possible to define functions using this coordinate system. For example, here is a spiral function:

It is also possible to find the area under a polar function, and that is what we will cover in this article.

We will look at how to calculate the area under a polar curve using three simple examples — a circle, and spiral, and a cardioid shape. We will also see how to calculate the area enclosed between two curves.

Polar functions

In Cartesian coordinates, it is common to define an equation y = f(x). We can plot the function f(x) by…