Rectangle method

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In integral calculus, the rectangle method uses an approximation to a definite integral, made by finding the area of a series of rectangles. In numerical computation, this has generally been superseded by more sophisticated methods of numerical integration.

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.