As the width of the boxes becomes narrower, the sum of their areas gets closer and closer to the area under the curve. Just like we saw in Introducing the Derivative, in the limit that the rectangles become infinitely thin (and thus of zero area themselves), we get the somewhat paradoxical result that their area sums to exactly to the area under the curve.

Integration is the mathematical process of accumulating these infinitesimally small areas, and the name for the (signed) area under the curve between two points \(x=a\) and \(x=b\) is the definite integral of the function \(f(x)\), written as $$A(x)=\int_a^bf(x)\,dx.$$

The activity below demonstrates how in the limit \(\Delta x \rightarrow 0\) the approximation to the area improves as the box width decreases.