What is a Riemann sum?
A sum of rectangle areas that approximates a definite integral.
Riemann Sum Calculator
Partition [a, b] into n equal subintervals and sum rectangle areas using left, right, or midpoints.
Set up the sum
Choose the rule and function, then enter a, b, and n.
About this calculator
How to use Riemann Sum Calculator, plus the formulas and logic behind the results.
Tool description
Riemann sums approximate definite integrals by summing f(xᵢ*) Δx over equal partitions.
How to use
- Choose left, right, or midpoint rule.
- Select f(x)=x² or a constant c.
- Enter interval [a, b] and positive integer n.
- Select Calculate to view the sum and Δx.
Formulas & reference
Δx = (b − a) / nLeft: Σ f(a + iΔx) Δx, i = 0..n−1Right: Σ f(a + (i+1)Δx) Δx, i = 0..n−1Mid: Σ f(a + (i+½)Δx) Δx, i = 0..n−1
Riemann tips
- n must be a positive integer
- Midpoint often converges faster than left/right
- Constant f gives exact area for any rule
- Compare with the exact integral of x² when possible
Calculation logic
Builds equal partitions and evaluates f at the sample points for the chosen rule.
Larger n generally improves the approximation for continuous f on [a, b].
Riemann Sum Calculator FAQ
Common questions about this calculator and how to use the results.
Left vs right vs mid?
They sample f at the left endpoint, right endpoint, or midpoint of each subinterval.
What is Δx?
The width of each equal subinterval: (b−a)/n.
Does n need to be even?
No for Riemann sums (unlike Simpson’s rule).
Can a > b?
Yes—Δx becomes negative and orientation is preserved.
Is this exact?
Only in special cases; otherwise it is an approximation.