Riemann Sum Calculator — FAQ

How does the Riemann sum calculator work?

Enter a function, the interval [a, b], and the number of subintervals n. The tool divides the interval into n equal parts of width Δx = (b−a)/n, evaluates the function at the appropriate sample points for the chosen method, and sums the rectangular (or trapezoidal) areas to approximate the area under the curve. The result is plotted with the shapes overlaid on the function graph.

Is this calculator free to use?

Yes, completely free with no sign-up, no downloads, and no limits. Run as many functions as you like.

What sum methods does it support?

It calculates left, right, midpoint, and trapezoidal Riemann sums simultaneously, so you can compare all four approximations for the same function and interval at a glance.

What is the difference between the methods?

The left and right sums use the function value at the left or right edge of each subinterval. The midpoint sum samples the function at the centre of each subinterval. The trapezoidal rule averages the left and right values, drawing trapezoids rather than rectangles. Midpoint and trapezoidal are typically more accurate than left or right sums for the same n, and the trapezoidal rule gives the exact answer for any linear function.

What functions can I enter?

You can enter standard math functions using the notation from math.js: polynomials (x^2, x^3 - 3*x), trigonometric (sin(x), cos(x), tan(x)), exponentials (e^x), logarithms (log(x)), square roots (sqrt(x)), and combinations (x * sin(x), e^(-x^2)).

Does it show the graph?

Yes. It plots the function curve and overlays the rectangles (for left, right, and midpoint methods) or trapezoids (for the trapezoidal rule) used in the approximation, so you can see exactly how the estimate is constructed. Changing the method updates the graph instantly.

What happens as I increase the number of subintervals?

The approximation generally converges toward the true definite integral. You can drag the slider or type a value for n and watch the rectangles narrow and the sum tighten in real time. For a monotone function, the left and right sums bracket the true value — as n → ∞, all four methods converge to the same limit.

Is my data stored anywhere?

No. Your function and inputs are processed entirely in your browser using math.js. Nothing is sent to a server, keeping your work completely private.

Does it work on mobile devices?

Yes. The tool is fully responsive — the graph scales to fill its container, and the layout stacks vertically on phones and tablets so inputs, graph, and results are all accessible without any horizontal scrolling.

How accurate is the result?

A Riemann sum is a numerical approximation, not the exact integral. Accuracy improves with more subintervals (larger n) and with the midpoint or trapezoidal method. The underlying arithmetic uses math.js floating-point evaluation, which is accurate to double precision for well-behaved functions.