Double Integral

Double Integrals

A double integral integrates a function of two variables over a 2D region:

$\iint_R f(x,y)\, dA$

Over a rectangular region $[x_a, x_b] \times [y_a, y_b]$.

The Midpoint Rule in 2D

Divide the region into $n_x \times n_y$ small rectangles. Evaluate $f$ at each midpoint:

$\iint f\, dA \approx \sum_i \sum_j f(x_i, y_j) \cdot \Delta x \cdot \Delta y$

Where $x_i = x_a + (i + 0.5) \Delta x$ and $y_j = y_a + (j + 0.5) \Delta y$.

Geometric Meaning

• $\iint 1\, dA$ = area of the region
• $\iint f\, dA$ = signed volume between $z = f(x,y)$ and the $xy$-plane

Examples

$\iint 1\, dA$ over $[0,3] \times [0,4]$ $= 12$ (area of $3 \times 4$ rectangle)

$\iint (x+y)\, dA$ over $[0,1] \times [0,1]$ $= 1$ (by exact integration: $\frac{1}{2} + \frac{1}{2}$)

$\iint x^2 y\, dA$ over $[0,2] \times [0,3]$ $= \frac{8}{3} \cdot \frac{9}{2} = 12$

Fubini's Theorem

For continuous $f$, you can integrate one variable at a time:

$\iint f\, dA = \int_{y_a}^{y_b} \left[ \int_{x_a}^{x_b} f(x,y)\, dx \right] dy$

Your Task

Implement double double_integral(double (*f)(double, double), double xa, double xb, double ya, double yb, int nx, int ny) using the 2D midpoint rule.