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Rectangle method

From Biocrawler, the free encyclopedia.

In mathematics, the rectangle method of integral calculus 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 rectangles that fill the space between two desired x-values.

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