Riemann Sum Calculator

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Approximate the area under a curve using Left, Right, or Midpoint Riemann sums.

Input Parameters

Start a

End b

Number of Subintervals (n)

Sum Type

Area Approximation

21.2800sq units

Approximate Area

Approximating ∫x² dx from 0 to 4

Δx (Width)0.4000

Intervals10

Visualization of Rectangles (f(x) = x²)

About this calculator

Quick Reference

Exact Integral1/3(b³-a³)

Error Term∝ 1/n

PrecisionHigh n = High Accuracy

What is a Riemann Sum?

A Riemann Sum is a method for approximating the total area under a curve (the definite integral) by dividing the region into vertical rectangles. As the number of rectangles increases (n → ∞), the sum approaches the exact value of the integral. This is the foundational concept of integral calculus.

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Pro Tips

Left Sum: Uses the left endpoint of each subinterval to determine height.
Right Sum: Uses the right endpoint, often overestimating increasing functions.
Midpoint: Usually the most accurate of the three basic Riemann sums.

Fun Facts

"Bernhard Riemann, a German mathematician, formalized this method in the mid-19th century."
"The symbol for integration (∫) is actually an elongated 'S', standing for 'summa' (sum)."

Formula: Area ≈ Σ f(xᵢ*)Δx