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Riemann sum convergence

Four methods of approximating a Riemann sum; Right, left, Maximum, and minimum. The values of the sums come closer together as the intervals get smaller.

A Riemann sum is a method of estimating the area underneath a curve by dividing it into rectangles or trapezoids, calculating the area of these shapes, then adding them together to get an approximation of area. The smaller the shapes are, the more accurate the estimation will be; the idea of using infinitely small rectangles is the basis of the definite integral. Riemann sums are commonly notated as

$ \begin{align}\sum_{k=1}^n f(x_k)\!\cdot\!\Delta x_k\end{align} $

with $ f(x_k)\!\cdot\!\Delta x_k $ being the area of a single rectangle. When $ n $ approaches infinity, the Riemann sum becomes a definite integral, which is notated as follows:

$ \begin{align}\lim_{n\to\infty}\sum_{k=1}^n f(x_k)\!\cdot\!\Delta x_k=\int\limits_a^b f(x)dx\end{align} $

where

$ \Delta x_k=\frac{b-a}{n}, x_k = a + k\Delta x_k $

See also

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