Area Between Curves

Area Between Two Curves

The area between two curves $f(x)$ and $g(x)$ on $[a, b]$ is:

$A = \int_a^b |f(x) - g(x)|\, dx$

The absolute value handles the case where the curves cross (one is above the other at different points).

When $f \geq g$ Everywhere

If $f(x) \geq g(x)$ throughout $[a, b]$, the area simplifies to:

$A = \int_a^b [f(x) - g(x)]\, dx = \int_a^b f(x)\, dx - \int_a^b g(x)\, dx$

Finding Intersection Points

To split the interval where the curves cross, find $x$ where $f(x) = g(x)$ — this is a root-finding problem.

Classic Example

Area between $y = x$ and $y = x^2$ on $[0, 1]$:

• $x \geq x^2$ on $[0, 1]$ since $x - x^2 = x(1-x) \geq 0$
• Area $= \int_0^1 (x - x^2)\, dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6} \approx 0.1667$

Numerically

Use the midpoint rule on $|f(x) - g(x)|$:

for (int i = 0; i < n; i++) {
    double x = a + (i + 0.5) * h;
    double diff = f(x) - g(x);
    sum += (diff < 0 ? -diff : diff);  /* abs */
}

Your Task

Implement double area_between(double (*f)(double), double (*g)(double), double a, double b, int n) that returns $\int_a^b |f(x) - g(x)|\, dx$.