Riemann Sum Calculator
Left, right, midpoint and trapezoid sums from your own function values.
Enter your values
ⓘ Live results
Results
- Trapezoid sum
- 22
- Left Riemann sum
- 14
- Right Riemann sum
- 30
- Average of left and right
- 22
- Subintervals n
- 4
- Δx
- 1
Quick answer
A Riemann sum approximates an integral as Δx times a sum of sampled heights; the left, right and midpoint rules differ only in where each rectangle's height is measured.
∫f dx ≈ Δx · Σ f(xᵢ)
How to use the Riemann Sum Calculator
- 1Enter the interval limits a and b.
- 2Enter the function value at every partition point, evenly spaced.
- 3Compare the left, right and trapezoid estimates — the true value sits between the left and right sums for a monotonic function.
Worked example
Using the values the calculator loads with:
Inputs
- Lower limit a0
- Upper limit b4
- Function values at each partition point (n+1 values)0, 1, 4, 9, 16
Results
- Trapezoid sum22
- Left Riemann sum14
- Right Riemann sum30
- Average of left and right22
- Subintervals n4
- Δx1
Frequently asked questions
Which rule is most accurate?
The trapezoid rule beats left or right sums for the same n, and Simpson's rule beats all three for smooth functions.
How many subintervals do I need?
Double n and see how much the answer moves. When the change is smaller than the precision you need, stop.
About this calculator
- Built and reviewed by James Whitfield · How we build and test calculators
- Report an error
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.