How to Approximate Area with Midpoint Sums

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 or right sum. The below figure shows why.

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.

For the three rectangles in the figure, the widths are 1 and the heights are f(0.5) = 1.25, f(1.5) = 3.25, and f(2.5) = 7.25. Add 1.25 + 3.25 + 7.25 to get the total area of 11.75.

The sums are heading toward 12, and if you could slice up the area into an infinite number of rectangles, you’d get the exact area of 12.

Here's the official midpoint rule: