Riemann Sum Calculator
Compute left, right and midpoint Riemann sums for a function over an interval.
Inputs
Integrand using x.
Midpoint sum
2.66000000
Left sum
2.28000000
Right sum
3.08000000
Width Δx
0.20000000
Step by step
Values used
f(x) = x^2; Lower bound (a) = 0; Upper bound (b) = 2; Rectangles (n) = 10
Left Riemann sum
L = Σ f(xᵢ)·Δx for i = 0 to n-1
Midpoint rule
M = Σ f((xᵢ + xᵢ₊₁)/2)·Δx
Midpoint sum
= 2.66000000
Left sum
= 2.28000000
Right sum
= 3.08000000
Width Δx
= 0.20000000
How it works
Riemann sums approximate ∫f(x)dx by dividing [a,b] into n equal rectangles. The left sum uses left endpoints, right sum uses right endpoints, and midpoint sum uses the center of each subinterval. The midpoint rule is generally more accurate (O(h²)) than left or right sums (O(h)).
Formulas
Left Riemann sum
L = Σ f(xᵢ)·Δx for i = 0 to n-1
- Δx
- (b-a)/n
- xᵢ
- a + i·Δx
Midpoint rule
M = Σ f((xᵢ + xᵢ₊₁)/2)·Δx
- Δx
- (b-a)/n
Frequently Asked Questions
Which Riemann sum is most accurate?
The midpoint sum is generally the most accurate, with error O(h²). Left and right sums have O(h) error but bound the integral from above/below for monotone functions.
How do Riemann sums relate to the integral?
The definite integral is defined as the limit of Riemann sums as n → ∞. Larger n gives better approximations.