Riemann Sum Demo

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Function Variables

Parent

Riemann Sum Variables

Type

# Rectangles

10

Lower Bound

-5.0

Upper Bound

5.0

Signed Area

Function

\[ f(x) = x \]

Riemann Sum Result

\[ \sum_{i=0}^{n} f(x_i)\Delta x = 0 \]

Result as # Rectangles -> Infinity (TRUE AREA)

Anti-Derivative of f(x)

\[ F(x) = \frac{x^2}{2} + C \]

Fundamental Theorem of Calculus

\[ F(b) - F(a) = \int_{a}^{b} f(x)\,dx \]

The Area Under the Curve

\[ \int_{a}^{b} f(x)\,dx = 0 \]

Guide

Riemann Sum Notes

What This Demo Shows

This graph displays a function on a coordinate grid. Right now you can switch between several function families with the dropdown. The next step will be adding rectangles, interval controls, and left/right/midpoint sum behavior.

Why Riemann Sums Matter

A Riemann sum approximates the area under a curve by splitting an interval into smaller pieces and adding the areas of rectangles. As the partitions get smaller, the approximation gets closer to the definite integral.

\[ \sum_{i=0}^{n-1} f(x_i)\Delta x \to \int_{a}^{b} f(x)\,dx \]