Finding the Area of a Region Bounded by a Single Polar Curve

In rectangular coordinates you slice a region into thin rectangles because y = f(x) gives a height at each x. Polar curves don't hand you a height, they hand you a distance from the origin at each angle. So the natural slice isn't a rectangle, it's a thin pie slice, a sector of angle d theta with radius r = f(theta). The area of a circular sector with angle d theta and radius r is (1/2) r² d theta, the same formula you'd use for a slice of pizza. Add up all those slices from theta = a to theta = b and you get A = (1/2) integral from a to b of r² d theta.

The hard part is usually not the formula, it's finding the right bounds. You need the exact range of theta that traces the region once, no more and no less. For a rose petal, that means finding the two consecutive angles where r = 0, since the petal starts and ends at the origin. For a cardioid or other closed curve that wraps once around, the bounds are often 0 to 2pi, but check that r doesn't retrace itself or go negative in a way that double-counts area.

This topic sits right after arc length in polar form and right before area between two polar curves. Once you can handle one curve's enclosed region, subtracting two such integrals to find area between curves is a small extension. The same sector-slicing idea also echoes the setup for polar arc length, where you integrate a different expression over the same kind of theta interval.