Subject: Calculus

Rectangle Method

Rectangular method computes an approximate of the numerical value of a definite integral by summing up the areas of the rectangles whose heights are determined by the values of a function which is illustrated by the figure below.

A definite integral is defined as a limit of Riemann sums, so any Riemann sum could be used as an approximation to the integral: If we divide [a, b] into subintervals of equal length \Delta x=\frac{b-a}{n} , then we have

\int_{a}^{b}f(x)dx \approx \sum_{i=1}^{n}f(x_{i}^{*})\Delta x

where x_i^* is any point in the i_th subinterval {x_{i-1}, x_i$}. If x_i^* is chosen to be the left endpoint of the interval, then x_i^*= x_i and we’ll have

\int_{a}^{b}f(x)dx\approx L_n \approx \sum_{i=1}^{n}f(x_{i-1})\Delta x

If f(x) \geq 0, then the integral represents an area and represents an approximation of this area by the rectangles shown in below and is called left endpoint approximation.

If we choose to be the right endpoint, then and we have

\int_{a}^{b}f(x)dx\approx R_n \approx \sum_{i=1}^{n}f(x_{i}\Delta x

This is called the right endpoint approximation which can be represented as

Another way that rectangle method can be used is considering the case where x_i^* is chosen to be the midpoint (\bar{x}_i) of the subinterval \left [x_{i-1}, x_i \right ].

We can clearly observe that the above figure appears to better than the R_n or L_n. For the midpoint approximation M_n, we approximate the values of the definite integral as follows

\int_{a}^{b}f(x)dx\approx M_n \approx \sum_{i=1}^{n}f(\bar{x}_{i}\Delta x

where

\Delta x =\frac{b-a}{n}

and

x_i=\frac{1}{2}\left (x_{i-1}+x_i\right )

is the midpoint of \left [x_{i-1},x_i \right ].

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