What is a Riemann sum?

A Riemann sum approximates the area under a curve by dividing it into rectangles. The sum of the rectangle areas approaches the exact definite integral as the number of rectangles increases.

What is the difference between left, right, and midpoint Riemann sums?

Left Riemann sums use the function value at the left edge of each rectangle, right sums use the right edge, and midpoint sums use the center. Midpoint typically converges fastest with error proportional to 1/n².

How many rectangles do you need for an accurate Riemann sum?

It depends on the function and method. For smooth functions like sin(x), 50 rectangles with the midpoint method often gives less than 0.1% error. The definite integral is the limit as the number of rectangles approaches infinity.

Riemann Sum Explorer

See how rectangles approximate the area under a curve. Increase the count and watch it converge to the exact integral.

Controls

Function

sin(x) x² x³ √x cos(x)+1 1/x

Rectangles (n)

Method

Left Right Midpoint

Results

n	10
Approx	-
Exact	-
Error	-

The Math Behind It

Riemann Sum Formula

Left sum: S = Σ f(xᵢ) Δx where xᵢ = a + i·Δx
Right sum: xᵢ = a + (i+1)·Δx
Midpoint: xᵢ = a + (i+½)·Δx
Δx = (b - a) / n

Convergence

As n → ∞, all three methods converge to the definite integral
Midpoint typically converges fastest (error ~ 1/n²)
Left/Right have error ~ 1/n for monotonic functions

Try This

Set n=1 and compare Left vs Right for sin(x) — which overestimates?
Increase n to 50 and see the error drop below 0.1%
Try x² with midpoint at n=4 — surprisingly accurate!
Watch the animation to see convergence in real-time

Key Insight

The definite integral ∫f(x)dx is defined as the limit of Riemann sums as the partition gets infinitely fine.

Riemann Sum Explorer

Controls

Results

The Math Behind It

Riemann Sum Formula

Convergence

Try This

Key Insight

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