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.
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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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.
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.
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.
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.
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.
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.
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.
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.