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

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

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

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

where

and

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

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