Rectangle method

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In integral calculus, the rectangle method (also called the Mid-Ordinate Rule) uses an approximation to a definite integral, made by finding the area of a series of rectangles.

Either the left or right corners, or top middle of the boxes lie on the graph of a function, with the bases run along the x-axis. The approximation is taken by adding up the areas (base multiplied by height, a function value) of the n rectangles that fill the space between two desired x-values.

$\int_a^b f(x)\,dx \approx \sum_{i=1}^{n} f(a+i'\Delta x)\Delta x \quad \mbox{ where } \Delta x = \frac{b-a}{n} \;,\; i' = \begin{cases} i-1 & \mbox{if left approx.}\\ i-\frac{1}{2} & \mbox{if midpoint approx.}\\ i & \mbox{if right approx.} \end{cases}$

The necessity of $a + i'Δ x$ arises when a is not zero, and as such the position of the first rectangle is not at $f (i'Δ x)$ but rather at $f (a + i'Δ x)$ . As n gets larger, the approximation gets more accurate. In fact, the limit of the approximation as n approaches infinity is exactly equal to the definite integral.

$\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(a+i'\Delta x)\Delta x$

This is true regardless of which i' is used. Although the midpoint approximation tends to be more accurate for finite n, the limit of all three approximations as n approaches infinity is the same, thus any of them can be used to calculate a definite integral.