# How do you use the midpoint rule to estimate area?

May 28, 2018

#### Explanation:

A good way to approximate areas with rectangles is to make each rectangle cross the curve at the midpoint of that rectangles top side.
A midpoint sum is a much better estimate of area than either a left-rectangle or right-rectangle sum.

The figure below shows you why it is better:

You can see in the figure that the part of each rectangle that’s above the curve looks about the same size as the gap between the rectangle and the curve.
A midpoint sum produces such a good estimate because these two errors roughly cancel out each other.

$\text{Midpoint Rectangle Rule}$

You can approximate the exact area under a curve between a and b

$A = {\int}_{a}^{b} y \cdot \mathrm{dx}$

with a sum of midpoint rectangles given by the following formula.

In general, the more rectangles, the better the estimate:

${A}_{\text{Midpoint}} = \frac{b - a}{n} \left[f \left(\frac{{x}_{0} + {x}_{1}}{2}\right) + f \left(\frac{{x}_{1} + {x}_{2}}{2}\right) + \ldots + f \left(\frac{{x}_{n} - 1 + {x}_{n}}{2}\right)\right]$

Where, n is the number of rectangles

$w i \mathrm{dt} h = \frac{b - a}{2}$

is the width of each rectangle, and the function values are the heights of the rectangles.