Riemann Sum

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Tap a term to see what it means, or .

Calculate

Example: the area under the curve y = x² between 0 and 2 is exactly 8/3 ≈ 2.667. With 8 rectangles, how close does it get? Change the function, the interval and the number of rectangles to work out your own case.

Function f(x) Start a End b Rectangles n Height taken

Step by step

What each term means

Definite Integral (∫ₐᵇ f(x) dx): The area under the curve of f between a and b. It is the value the sum of the rectangles approximates, and reaches as n goes to infinity.
Sum of the Rectangles (Σ): Adds the areas of the n rectangles that cover the interval, from i = 1 to i = n.
Height (f(xᵢ*)): The value of the function at a point xᵢ* chosen inside each part: the left edge, the right edge or the midpoint. The midpoint usually errs much less.
Width (Δx): The width of each rectangle: (b − a) / n. The more rectangles, the narrower they are and the tighter the approximation.