Area with Polar Coordinates

Polar area is simply the amount of space enclosed by a polar curve, measured by cutting the region into thin pizza slices that fan out from the origin and adding up their areas. Each slice is a sector, the mathematician's word for a pie-slice-shaped piece of a circle. It's good for regions that wrap around a center, like flower petals, a cardioid, the patch a windshield wiper cleans or the coverage of an antenna. The formula $\int \tfrac12 r^2\,d\theta$ looks odd at first, but it's just three small pieces: $r$ (how far out the edge is at that angle), $d\theta$ (how wide the slice is, as an angle) and $\tfrac12$ (because a slice is a triangle-ish wedge, not a full rectangle). Multiply them for one slice and the integral adds up all the slices as you turn; the rest is just choosing where to start and stop.

Why not rectangles? For $y = f(x)$ we sliced area into vertical rectangles, because the curve was a height above the x-axis. A polar curve isn't a height above anything. It's a distance from the origin at each angle. So the natural slice isn't a rectangle; it's a thin wedge, like a slice of pizza, with its point at the origin.

Where does $\tfrac12 r^2\,d\theta$ come from? A full circle of radius $r$ has area $\pi r^2$ and angle $2\pi$. A sector with angle $\Delta\theta$ is the fraction $\frac{\Delta\theta}{2\pi}$ of that circle, so its area is $\frac{\Delta\theta}{2\pi}\cdot\pi r^2 = \tfrac12 r^2\Delta\theta$. Approximate the region by thin sectors, let $\Delta\theta \to 0$, and the sum becomes the integral, the same way Riemann sums did.

Choosing limits is the real work. Sketch the curve and find the angles where it starts and ends a loop, often where $r = 0$. Use symmetry whenever you can: compute one petal and multiply.

Between two curves, subtract the inner radius squared from the outer:

with the limits at the angles where the curves cross. Think about why: it's the big slice minus the small slice, and each slice's area has its own $r^2$ in it.

A common mistake. It's tempting to subtract the radii first and then square, $\left(r_{\text{outer}} - r_{\text{inner}}\right)^2$, because with rectangles you'd subtract the heights first. It isn't the same thing. The area of each slice depends on $r^2$, so you have to square each radius before you subtract.

A windshield wiper sweeps a polar region, and its cleaned area is exactly this integral. Radar coverage maps, sprinkler patterns and the areas swept by orbiting planets (Kepler's second law) are polar areas too.