Mathematics

The key insight is that polar coordinates naturally sweep out sectors of area, not rectangles. Instead of slicing vertically (as in Cartesian coordinates), you slice by angle, and each thin slice is approximately a circular sector with radius $r = f(\theta)$ and angular width $d\theta$. The area of such a sector is $\frac{1}{2}r^2 \, d\theta$, so you integrate this expression across your angular interval to capture the total enclosed area.

The key insight is that polar coordinates naturally sweep out sectors of area, not rectangles. Instead of slicing vertically (as in Cartesian coordinates), you slice by angle, and each thin slice is approximately a circular sector with radius $r = f(\theta)$ and angular width $d\theta$. The area of such a sector is $\frac{1}{2}r^2 \, d\theta$, so you integrate this expression across your angular interval to capture the total enclosed area.