Trapezoidal Rule

Trapezoidal Rule

Instead of rectangles, use trapezoids — connect adjacent points with line segments:

$\frac{h}{2} \cdot \left[f(a) + 2f(a+h) + 2f(a+2h) + \cdots + 2f(b-h) + f(b)\right]$

where $h = \frac{b-a}{n}$.

Derivation

Each trapezoid has two parallel sides $f(x_i)$ and $f(x_{i+1})$ and width $h$. Its area is $\frac{h \cdot (f(x_i) + f(x_{i+1}))}{2}$. Sum all $n$ trapezoids — the interior points appear twice:

$\text{Area} = \frac{h}{2} \cdot \left[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\right]$

Accuracy

Error is $O(h^2)$ — halving $n$ reduces error by factor of 4. The trapezoidal rule is exact for linear functions.

Error Formula

$\text{Error} \leq \frac{(b-a)^3}{12n^2} \cdot \max|f''(x)|$

For $f(x) = \sin(x)$, $\max|f''| = 1$, so on $[0, \pi]$ with $n=100$: error $\leq \frac{\pi^3}{120000} \approx 0.000258$.

Comparison

Your Task

Implement double trapezoid(double (*f)(double), double a, double b, int n).