Riemann Sum Calculator

Enter a function f(x), set your lower limit (a) and upper limit (b), choose the number of subintervals (n), and select a Riemann sum type (Left, Right, Midpoint, or Trapezoidal). You get back the approximate definite integral value, the subinterval width Δx, and a bar chart visualizing the rectangles (or trapezoids) used in the approximation. Also try the calculate y(x) at Evaluated Point, Solution Type & Integration Constant C — Separable Differential Equation.

Use JavaScript math syntax: x^2 → x*x, sqrt(x) → Math.sqrt(x), sin(x) → Math.sin(x)

More subintervals give a more accurate approximation (max 200).

Results

Approximate Integral Value

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Subinterval Width (Δx)

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Number of Subintervals Used

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Method

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Results Table

Frequently Asked Questions

What is a Riemann sum?

A Riemann sum is a method for approximating the definite integral of a function over an interval by dividing the interval into subintervals and summing the areas of rectangles (or trapezoids) built on each subinterval. As the number of subintervals increases, the approximation converges to the exact integral. See also our Differential Equation Solver.

What is the difference between Left, Right, and Midpoint Riemann sums?

The Left Riemann sum uses the left endpoint of each subinterval to determine rectangle height. The Right Riemann sum uses the right endpoint. The Midpoint Rule uses the center of each subinterval. For monotonically increasing functions, Left underestimates and Right overestimates the true integral; Midpoint is often the most accurate of the three.

How does the Trapezoidal Rule differ from Riemann sums?

The Trapezoidal Rule approximates each strip as a trapezoid rather than a rectangle, using both the left and right endpoint heights averaged together. This generally produces a more accurate approximation than the simple left or right Riemann sums with the same number of subintervals.

How do I enter my function correctly?

Use standard JavaScript math syntax: for x², type x*x; for √x, type Math.sqrt(x); for sin(x), type Math.sin(x); for eˣ, type Math.exp(x); for |x|, type Math.abs(x). Constants like π can be entered as Math.PI. You might also find our find Definite Integral ∫f(x)dx with Definite Integral Calculator useful.

How many subintervals should I use for accuracy?

More subintervals produce a more accurate approximation. For most classroom problems, n = 4 to 20 is common. For higher precision, use n = 100 or more. The Trapezoidal and Midpoint rules converge faster than Left/Right sums, so they need fewer subintervals for the same accuracy.

What is Δx (delta x) in the Riemann sum formula?

Δx is the width of each subinterval, calculated as (b − a) / n, where a and b are the lower and upper limits and n is the number of subintervals. All rectangles have this same width; only their heights (function values) differ.

Can a Riemann sum give a negative result?

Yes. If the function has regions below the x-axis on [a, b], those rectangles contribute negative area. The Riemann sum computes the net signed area, which can be negative, zero, or positive depending on the function.

What is the formula for the Left Riemann sum?

The Left Riemann sum is: Σ f(xᵢ) · Δx for i = 0 to n−1, where xᵢ = a + i·Δx and Δx = (b−a)/n. The Right sum uses i = 1 to n (right endpoints), while the Midpoint uses x = a + (i + 0.5)·Δx as the sample point.