Approximating Areas with Riemann Sums

A curve doesn't have straight edges, so you can't find the area under it with a simple length times width formula. Rectangles do. So you chop the interval into equal pieces, build a rectangle over each piece using the function's height at some sample point (left edge, right edge, or midpoint), and add up all those rectangle areas. That sum is only an estimate, since each rectangle either overshoots or undershoots the curve, but it's an estimate you can actually compute.

The fix for that overshoot and undershoot is to use more rectangles. Thinner rectangles hug the curve more closely, so the gaps between the rectangle tops and the curve shrink. As the number of rectangles n grows, the width of each one, b/n, shrinks toward zero, and the total error shrinks with it. Push n toward infinity and the sum stops being an approximation at all: it becomes the exact area, which is the definite integral.

This is why the definite integral is defined as the limit of a Riemann sum as n goes to infinity. Every definite integral you compute later, using the Fundamental Theorem of Calculus, is secretly standing in for this infinite sum of infinitely thin rectangles. Riemann sums are also how you'll approximate area from a table of data, where no formula for f(x) exists and antiderivatives aren't an option.