Polar Area

Polar Area

In polar coordinates, a curve is defined by radius as a function of angle: $r = r( heta)$.

The area enclosed by a polar curve from $heta = a$ to $heta = b$:

A = rac{1}{2} int_a^b [r( heta)]^2 , d heta

Why the Formula Works

Each infinitesimal slice at angle $heta$ is a sector with radius $r( heta)$ and angle $d heta$. A sector's area is rac{1}{2}r^2 d heta (fraction of circle area $pi r^2$).

Classic Examples

Full circle $r = R$ on $[0, 2pi]$:

A = rac{1}{2} int_0^{2pi} R^2 , d heta = rac{1}{2} cdot R^2 cdot 2pi = pi R^2

Semicircle $r = 1$ on $[0, pi]$:

A = rac{1}{2} int_0^{pi} 1 , d heta = rac{pi}{2} approx 1.5708

Rose curves and limaçons: more complex integrands requiring numerical integration.

Numerical Approach

Apply the midpoint rule:

#define PI 3.14159265358979
double h = (b - a) / n;
for each midpoint θ:
    double r = r_fn(θ);
    sum += r * r;
return 0.5 * sum * h;

Your Task

Implement double polar_area(double (*r_fn)(double), double a, double b, int n).