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Georg Friedrich Bernhard Riemann (1826-1866) is the person who formulated the modern definition of an integral. He decided that it was not absolutely necessary that each rectangle have the same width. They do not even need to be the same type (i.e. they all do not need to be endpoint or midpoint rectangles).

Examine the rectangles used at right to estimate the area under a curve. Will they still give a good estimate of area even though the rectangles do not have the same width?

While the rectangles in the diagram will give a good estimate, discuss some of the advantages using rectangles that have the same width?

First & second quadrants, increasing cubic coming from bottom left of negative 5, changing from concave down to concave up at about (0, comma 5.5), continuing through the point (4, comma 8), with 4 shaded rectangles between the curve & x axis, from negative 4 to 0, with midpoint of top edge on the curve, & 9 unequal width rectangles between the curve & x axis, from 0 to 5, with the skinnier ones having top right edge on the curve, & wider ones have midpoint on the curve.