Left Riemann Sum

The Definite Integral

The definite integral $\int_a^b f(x)\, dx$ is the signed area between $f$ and the x-axis from $a$ to $b$.

It is defined as the limit of Riemann sums — sums of rectangle areas:

$\int_a^b f(x)\, dx = \lim_{n \to \infty} \sum_{i} f(x_i) \cdot \Delta x$

where $\Delta x = \frac{b-a}{n}$ and $x_i$ is some point in the $i$-th subinterval.

Left Riemann Sum

Use the left endpoint of each subinterval:

$\sum_{i=0}^{n-1} f(a + i \cdot h) \cdot h \quad \text{where } h = \frac{b-a}{n}$

a    a+h  a+2h       b
|----|----|----...---|
^    ^    ^
left endpoints used

Accuracy

For an increasing function, the left sum underestimates the integral (the rectangles don't reach the curve). For a decreasing function, it overestimates. Error is $O(h)$ — halving $n$ halves the error.

Example

$\int_0^1 x^2\, dx$ with $n=4$, $h=0.25$:

• $0.25 \cdot (f(0) + f(0.25) + f(0.5) + f(0.75))$
• $= 0.25 \cdot (0 + 0.0625 + 0.25 + 0.5625)$
• $= 0.25 \cdot 0.875 = 0.21875$
• Exact: $\frac{1}{3} \approx 0.3333$ — the left sum underestimates

Your Task

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