How to Approximate Area with Midpoint Rectangles

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.

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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

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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,

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with a sum of midpoint rectangles given by the following formula. In general, the more rectangles, the better the estimate:

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Where, n is the number of rectangles,

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

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