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Area inside a polar curve

Area Inside a Polar Curve

To understand the area inside of a polar curve

$r=f(\theta)$ , we start with the area of a slice of pie. If the slice has angle

$\theta$ and radius

$r$ , then it is a fraction

$\frac{\theta}{2\pi}$ of the entire pie. So its area is

$\frac{\theta}{2\pi} \pi r^2 = \frac{r^2}{2}\theta.$

Now we can compute the area inside of polar curve $r=f(\theta)$ between angles $\theta=a$ and $\theta=b$ . As with all bulk quantities, we

In our case, the pieces are slices of angle

$\Delta \theta = (b-a)/N$ . These aren't exactly pie slices, since the radius isn't constant, but it's a good approximation when

$N$ is large and

$\Delta \theta$ is small. The

$i$ -th slice has area approximately

$f(\theta_i^*)^2 \Delta \theta/2$ , where

$\theta_i^*$ is a representative angle between

$a+(i-1)\Delta \theta$ and

$a+i \Delta \theta$ , so the whole thing has area approximately

$\displaystyle{\sum_{i=1}^N \frac{f(\theta_i^*)^2}{2}\Delta \theta}$ . Taking a limit as

$N \to \infty$ gives the integral

$\int_a^b \frac{f(\theta)^2}{2} d\theta, \qquad \hbox{or equivalently} \qquad \frac{1}{2}\int_a^b f(\theta)^2 d\theta.$

The following video goes over the derivation of this formula, and uses it to compute the area inside one lobe of a cardioid.