★☆☆ Easy

∫ Riemann Integral

Choose a function and method, drag the interval and subdivisions slider — watch the shaded bars home in on the exact area under the curve.

Function

a = 0.0

b = 3.0

n = 10 subdivisions

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Approximation

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Exact integral

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Error

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Riemann Sums

A Riemann sum approximates the area under f(x) by summing the areas of n rectangles (or trapezoids) of equal width Δx = (b−a)/n. As n → ∞ the sum converges to the exact definite integral ∫ₐᵇ f(x) dx.

Left rule: rectangle height = f(xᵢ) | Right rule: f(xᵢ₊₁) | Midpoint: f(xᵢ + Δx/2) | Trapezoid: [f(xᵢ)+f(xᵢ₊₁)]/2 | Simpson: [f(xᵢ)+4f(mid)+f(xᵢ₊₁)]/6 per pair.