Riemann Sums – AP Calculus AB

Learning Objectives

Approximate definite integrals using Riemann sums

Compare left, right, midpoint, and trapezoidal methods

Understand that as n→∞, all methods converge to the exact integral

Identify over- and under-estimates for increasing/decreasing functions

Key Equations

\(\displaystyle \sum_{i=1}^{n} f(x_i^*)\Delta x \approx \int_a^b f(x)\,dx\)

\(\Delta x = \frac{b-a}{n}\)

\(\text{Left: } x_i^* = a + (i-1)\Delta x\)

\(\text{Right: } x_i^* = a + i\Delta x\)

\(\text{Mid: } x_i^* = a + (i-\tfrac{1}{2})\Delta x\)

Why It Matters

Riemann sums are the foundation of integral calculus — they show how adding up infinitely many infinitesimal rectangles gives the exact area under a curve. This same idea powers numerical integration in science, engineering, and computer graphics.