Riemann Sum

A Riemann sum approximates the area under a curve $y = f(x)$ on $[a, b]$ by dividing the interval into $n$ subintervals of width $\Delta x = (b-a)/n$ and summing the areas of $n$ rectangles:

$S_n = \sum_{i=1}^n f(x_i^*) \, \Delta x$

where $x_i^*$ is a sample point in the $i$-th subinterval. Common choices:

• Left Riemann sum: $x_i^* = a + (i-1)\Delta x$.
• Right Riemann sum: $x_i^* = a + i \Delta x$.
• Midpoint rule: midpoint of subinterval (more accurate).

As $n \to \infty$ (rectangles get arbitrarily thin), if $f$ is integrable, the Riemann sum converges to the definite integral:

$\int_a^b f(x)\,dx = \lim_{n \to \infty} S_n.$

This definition of the integral ties together discrete summation and continuous area, motivating the integral notation $\int$ as a "stretched S" for sum. Riemann sums also underlie all numerical integration (trapezoidal rule, Simpson's rule).