Riemann Sum

Approximate area with left, right, midpoint, or trapezoid rectangles

About this calculator

The Riemann Sum Calculator approximates the definite integral ∫ₐᵇ f(x) dx with up to 500 subintervals using the left, right, midpoint, or trapezoid rule, drawing every rectangle over the curve. It reports the approximation, the subinterval width Δx, a high-resolution Simpson's-rule reference value, and the absolute error, so you can watch the sum converge as n grows or the method changes. Functions undefined anywhere on the sample grid are flagged instead of summed. Ideal for learning how integration emerges from finite sums.

How to use the Riemann Sum calculator

  1. Enter the values for your problem into the input fields.
  2. Read the result — it updates instantly as you type.
  3. Check the formula and the visual explanation to follow how the answer was found.
  4. Copy the page URL to share the exact calculation.

Common examples

  • x² on [0, 3] with 6 left rectangles → 6.875 vs exact 9 (error 2.125, Δx = 0.5)
  • Same integral with the midpoint rule → 8.9375 (error only 0.0625)
  • Same integral with the trapezoid rule → 9.125 (error 0.125)
  • sin(x) on [0, π] with 100 midpoint bars → 2.00008224907 vs exact 2

Frequently asked questions

How does the Riemann Sum calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the Riemann Sum calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.