Riemann Sum Calculator
Left, right, midpoint and trapezoid sums from your own function values.
Enter your values
Enter the lower limit a you are working with.
Enter the upper limit b you are working with.
Enter the function values at each partition point (n+1 values) you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- 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)
At a glance
| What it does | Left, right, midpoint and trapezoid sums from your own function values. |
|---|---|
| Category | Math |
| Inputs needed | Lower limit a, Upper limit b, Function values at each partition point (n+1 values) |
| Main output | Trapezoid sum |
| Formula | ∫f dx ≈ Δx · Σ f(xᵢ) |
| Cost | Free — no sign-up, no download |
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.
Inputs explained
- Lower limit a
- Enter the lower limit a you are working with.
- Upper limit b
- Enter the upper limit b you are working with.
- Function values at each partition point (n+1 values)
- Enter the function values at each partition point (n+1 values) you are working with.
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.
What do I need to enter into the Riemann Sum Calculator?
Just 3 values: lower limit a, upper limit b, function values at each partition point (n+1 values). Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Riemann Sum Calculator work out the 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. It applies the formula ∫f dx ≈ Δx · Σ f(xᵢ) and shows the working so you can check each step by hand.
Is the Riemann Sum Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.