Polar Double Integral

Integration in Polar Coordinates

Some regions are naturally described in polar coordinates $(r, \theta)$ rather than Cartesian $(x, y)$.

The Polar Jacobian

When converting to polar coordinates $x = r\cos(\theta)$, $y = r\sin(\theta)$, the area element becomes:

$dA = r\, dr\, d\theta$

The extra $r$ factor (the Jacobian) accounts for the stretching near the origin.

The Formula

$\iint_R f(x,y)\, dA = \int_a^b \int_{r_0}^{r_1} f(r\cos\theta,\, r\sin\theta) \cdot r\, dr\, d\theta$

Midpoint Rule in Polar Coordinates

Divide $[r_0, r_1]$ into $n_r$ strips and $[a, b]$ into $n_\theta$ angular sectors:

$\approx \sum_i \sum_j f(r_i \cos\theta_j,\, r_i \sin\theta_j) \cdot r_i \cdot \Delta r \cdot \Delta\theta$

Classic Example: Area of a Disk

For a disk of radius $R$:

$\int_0^{2\pi} \int_0^R r\, dr\, d\theta = 2\pi \cdot \frac{R^2}{2} = \pi R^2 \checkmark$

Applications

• Area of circles, sectors, rings
• Volume of cylinders and cones
• Integrals with $x^2 + y^2$ in them (naturally round)

Your Task

Implement double polar_double_integral(double (*f)(double, double), double r0, double r1, double a, double b, int nr, int ntheta).

The function $f$ takes Cartesian $(x,y)$ coordinates. Inside, convert $(r, \theta)$ to $(x, y)$ and include the $r$ factor.

Use #include <math.h> for cos and sin.