A good way to approximate areas with rectangles is to make each rectangle cross the curve at the midpoint of that rectangle's top side. A midpoint sum is a much better estimate of area than either a left-rectangle or right-rectangle sum. The below figure shows why.
![image0.jpg](https://www.dummies.com/wp-content/uploads/202655.image0.jpg)
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.
The figure above shows how you’d use three midpoint rectangles to estimate the area under
![image1.png](https://www.dummies.com/wp-content/uploads/202656.image1.png)
from 0 to 3. For the three rectangles, their widths are 1 and their heights are f(0.5) = 1.25, f(1.5) = 3.25, and f(2.5) = 7.25. Area = base x height, so add 1.25 + 3.25 + 7.25 to get the total area of 11.75.
Using the definite integral, you find that the exact area under this curve turns out to be 12, so the error with this three-midpoint-rectangles estimate is 0.25. Contrast that with the much worse errors of the three-left-rectangles estimate and the three-right-rectangles estimate of 4.0 and 5.0, respectively.
Here's the official midpoint rule:
Midpoint Rectangle Rule—You can approximate the exact area under a curve between a and b,
![image2.png](https://www.dummies.com/wp-content/uploads/202657.image2.png)
with a sum of midpoint rectangles given by the following formula. In general, the more rectangles, the better the estimate:
![image3.png](https://www.dummies.com/wp-content/uploads/202658.image3.png)
Where, n is the number of rectangles,
![image4.png](https://www.dummies.com/wp-content/uploads/202659.image4.png)
is the width of each rectangle, and the function values are the heights of the rectangles.