What is a Riemann sum?

A Riemann sum approximates the area under a curve by dividing the interval into n equal subintervals, drawing a rectangle in each, and summing their areas. The height of each rectangle comes from the function value at the left endpoint, right endpoint, or midpoint of the subinterval.

Which method is most accurate?

The midpoint rule is generally the most accurate of the three for smooth functions. The left and right sums tend to over- or under-estimate depending on whether the function is increasing or decreasing. All three converge to the exact integral as n approaches infinity.

What functions can I enter?

You can use standard math notation: x^2, x^3, sin(x), cos(x), tan(x), sqrt(x), abs(x), ln(x), log(x), exp(x), and combinations like 3*x^2 + 2*x - 1. Use * for multiplication.

Why does my result differ from the exact integral?

Riemann sums are approximations. The error decreases as n increases. For a highly curved function with a small n, the rectangles miss parts of the curve. Use n = 100 or more for close approximations.

What is the difference between left and right Riemann sums?

The left sum uses the function value at the left edge of each subinterval; the right sum uses the right edge. For an increasing function, the left sum underestimates and the right sum overestimates the true integral.

Riemann Sum Calculator

Name: Riemann Sum Calculator
Availability: InStock
Author: Nham Vu

Enter a function, interval, and number of subintervals to compute left, right, or midpoint Riemann sum approximations with a visual graph.

Settings

f(x)

Use * for multiply, ^ for power

Lower bound (a)

Upper bound (b)

Subintervals (n)

Method

Left Mid Right

Graph

Summary

Enter a function, interval, and number of subintervals to compute left, right, or midpoint Riemann sum approximations with a visual graph.

How it works

Enter a mathematical function using standard notation (e.g. x^2, sin(x), sqrt(x)).
Set the lower and upper bounds of the integration interval.
Choose the number of subintervals n (more subintervals = better accuracy).
Select left, right, or midpoint Riemann sum method.
The calculator divides the interval into n equal subintervals and sums rectangle areas.
A graph shows the function curve and the Riemann rectangles.

Use cases

Verify hand-calculated Riemann sum answers for calculus homework.
Visualize how rectangle approximations converge to the true integral as n increases.
Compare accuracy of left, right, and midpoint methods for a given function.
Explore numerical integration concepts in introductory calculus courses.
Estimate areas under curves that lack closed-form antiderivatives.
Check understanding before taking a calculus exam.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu