Riemann Sum Approx – AP Calculus AB

Learning Objectives

The Summation $\Sigma$: The area under a curve is approximated by slicing it into $n$ geometric shapes (rectangles or trapezoids) of width $\Delta x = \frac{b-a}{n}$.

Under/Over Estimates: Depending on whether the function is strictly increasing or decreasing, an LRAM or RRAM will systematically over- or under-estimate the true area.

The Integral Limit: As $n \to \infty$, the width $\Delta x \to 0$ ($dx$). The clunky Riemann Sum mathematically transforms into the exact Definite Integral $\int_a^b f(x)dx$.

Tags

IntegrationRiemann SumsLRAMRRAMTrapezoid

       FUNCTION TO INTEGRATE

       $f(x) = -\frac{1}{2}x^2 + 5$

Approximation Method

Subdivision Control

Partitions (n) = 4

Area Evaluation $[a,b] = [0, 3]$

Base $\Delta x$ 0.75

SERIES APPROXIMATION $\sum Area_i$

Estimated Area: 0.00

TRUE DEFINITE INTEGRAL

True Exact Area: 10.50

Error: 0.00

LRAM Overestimate: Because $f(x)$ is decreasing over $[0,3]$, taking the height from the LEFT side of the rectangle forces it to poke out above the curve, resulting in an OVERESTIMATE.