Midpoint Rule

Midpoint Rule

Use the midpoint of each subinterval:

$\sum_{i=0}^{n-1} f\!\left(a + \left(i + 0.5\right) h\right) \cdot h$

a   a+h  a+2h        b
|----|----|----...----|
 ^    ^    ^
 midpoints used

Why Midpoints Are Better

The midpoint rule has $O(h^2)$ error — the same order as the trapezoidal rule, and better than left/right ($O(h)$). Intuitively, the midpoint is the "best representative" of a subinterval because it cancels out the linear error terms from both sides.

Comparison for $\int_0^1 x^2\, dx = \frac{1}{3}$

The midpoint rule is more accurate than both left and right for the same $n$.

Exact for Linear Functions

The midpoint rule integrates linear functions exactly, regardless of $n$. For $f(x) = 2x + 1$ on $[0, 3]$:

• $\int_0^3 (2x+1)\, dx = \left[x^2+x\right]_0^3 = 12$
• Midpoint rule with any $n$: $\sum f(a + (i+0.5)h) \cdot h = 12$ ✓

Your Task

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