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To estimate the area under a curve, rectangles are often the easiest shape to use. However, there are different ways to choose the heights of the rectangles. You have already used left endpoint and right endpoint rectangles. Another way is to use midpoint rectangles, which have heights defined at the midpoints of the intervals. For example, for the function graphed at right, the first midpoint rectangle has a height of .

Calculate the height of the other two rectangles then use them to approximate the area under the curve for .

Use your calculator to find the values of and .

The base of each rectangle is . The approximation for the area under the curve is .

First quadrant graph, x axis scaled in ones from 0 to 4, with curve of the function, f of x = 1 half x + cosine x, & 3  vertical shaded bars, bottom edges on the x axis, each with width of 1, starting at x = 1, with top edge midpoint of each bar, on the curve.