Riemann Sum Visualizer Math

What is a Riemann Sum?

Quick Answer: A Riemann sum approximates the area under a curve f(x) by dividing the interval into rectangles. You choose the number of rectangles and whether their height is taken from the left, right, or midpoint of each subinterval. As the number of rectangles increases, the sum approaches the definite integral.

Theory Behind Riemann Sums

The definite integral of f(x) from a to b is the limit of Riemann sums as the number of subdivisions n → ∞. The tool evaluates f(x) using Math.js. For each subinterval of width Δx = (b-a)/n, the rectangle height is f(x_i), where x_i is the left, right, or midpoint. The sum = Σ f(x_i) Δx. The total shaded area represents this sum. The exact integral (if calculable analytically) isn't shown; the focus is on the geometric interpretation.

Step-by-Step Examples

Example 1: Quadratic Area

1. Use f(x)=x^2 on [0,3] with left endpoints and n=10. The sum underestimates the area because f is increasing.
2. Switch to right endpoints; the sum overestimates. Midpoint gives a better approximation.
3. Increase n to 50 to see rectangles become thinner and the sum stabilises around 9.

Example 2: Sine Wave

1. Enter f(x)=sin(x), interval [0, π] (~3.14). Use left sum with n=10.
2. Observe rectangles above and below the x-axis. The sum approximates the net area (2).

Example 3: Exponential Growth

1. Set f(x)=exp(x), a=0, b=2. Compare left and right sums with n=5 to see the gap narrow with larger n.

Frequently Asked Questions