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).

For example, examine the rectangles at right to estimate the area under $f(x)$. Will they still give a good estimate of area even though the rectangles do not have the same width?