Type analytic expressions such as sin(x), exp(-x^2), x^3 - 2x, or combinations that call ln, abs, sgn, and constants like pi.
Need a quick class demo? Toggle the fill option to emphasise signed area, compare how each rule converges as n grows, and use the CSV or LaTeX buttons to build worksheets instantly.
Inputs & options
Visualisation
Results
How it's calculated
Interpretation & worked example
A Riemann sum approximates the definite integral by adding up small “area” pieces: Sₙ = Σ f(xᵢ*)·Δx, where Δx = (b−a)/n. With more subintervals (larger n), the approximation usually improves.
Quick intuition (when it over/under-shoots)
- If f(x) is increasing on [a,b], the left sum tends to underestimate and the right sum tends to overestimate.
- Midpoint and trapezoid are often much closer for smooth functions.
- Simpson can be very accurate on smooth curves, but it requires an even n.
Mini example: f(x)=x² on [0,1] with n=4
Here Δx = 0.25 and the true value is ∫₀¹ x² dx = 1/3 ≈ 0.3333.
- Left: 0.21875 (undershoots)
- Right: 0.46875 (overshoots)
- Midpoint: 0.328125 (close)
- Trapezoid: 0.34375 (close)
- Simpson (n=4): 0.33333… (exact for this polynomial)
Tip: use the Rule dropdown to compare methods at the same n, then increase n to see convergence.
References
How to use this calculator effectively
This guide helps you use Riemann Sums Explorer in a repeatable way: set a baseline, change one variable at a time, and interpret the output with clear assumptions before sharing or exporting results.
How it works
The calculator takes your input values, applies a deterministic formula set, and returns output using display rounding only at the final step. This means the tool is best used as a comparison engine: keep one scenario as a reference, then test alternate assumptions so you can quantify how sensitive the final answer is to each input.
When to use
Use this page when you need a fast planning estimate, a classroom sanity check, or a shareable scenario that another person can reproduce from the same parameters. It is especially useful before deeper modeling, because it exposes direction and magnitude quickly without requiring sign-in or setup friction.
Common mistakes to avoid
- Mixing units (for example, percent vs decimal, or monthly vs yearly assumptions).
- Changing multiple fields at once, which makes it hard to explain why results shifted.
- Comparing outputs from different tools without aligning defaults and conventions.
- Reading rounded display numbers as exact values in downstream calculations.
Interpretation and worked example
Run a baseline case first and keep a copy of that output. Next, change one assumption to represent your realistic alternative, then compare the delta in both absolute and percentage terms. If the direction matches your domain intuition and the size of change is plausible, your setup is likely coherent. If not, review units, sign conventions, and hidden defaults before drawing conclusions.
See also
FAQ
Which Riemann sum rule should I pick?
Left/right sums follow the orientation of rectangles, which is great for quick estimates but can over- or undershoot when f is monotone. The trapezoid rule is second-order accurate and balances speed with precision. Simpson's rule reaches fourth-order accuracy on smooth functions, while midpoint splits the difference with a symmetric single-rectangle view.
Why must Simpson's rule use an even n?
Simpson's rule joins point triplets with quadratics, so the interval must break into an even number of subintervals. The explorer automatically increases n by 1 when needed and highlights the adjustment in the steps.
How many subintervals n should I use?
For smooth functions start with n = 50–100 and increase n to reduce error. Simpson converges faster but requires even n; the tool adjusts n automatically when needed.
Can I enter piecewise or non‑analytic functions?
Yes. You can use abs, sgn, and constants like pi. Non‑finite evaluations are skipped and treated as zero; the step log notes any skipped points.
What should I enter first?
Start with the minimum required inputs shown above the calculate button, then keep optional settings at their defaults for a first run. After you get a baseline result, change one parameter at a time so you can see exactly what caused the output to move.