5.1: Area Between Curves: Integrating With Respect to x

Consider two continuous functions defined on a closed interval from a to b. The region between these curves is bounded vertically by their graphs and horizontally by the endpoints of the interval. The objective is to measure the area of this region.

An initial estimate of the area can be obtained by dividing the interval into a large number of narrow vertical strips of equal width. Each strip is approximated by a rectangle whose height is given by the vertical difference between the two functions at a selected point within the strip. Adding the areas of these rectangles produces a Riemann sum approximation of the region. This process is represented by

Because the rectangles do not exactly match the curved boundaries, this sum provides only an approximation when a finite number of strips is used. However, as the number of strips increases and their width decreases, the approximation improves. In the limiting case, the rectangles become infinitely thin, and the summation converges to the exact area.

This limiting process replaces the Riemann sum with a definite integral, which computes the area using infinitely thin vertical slices. The exact area between the curves is given by

This method of finding the area between curves has important applications beyond pure mathematics. In economics, for example, the area between the line of perfect equality and the curve representing actual income distribution is used to quantify income inequality. The magnitude of this area provides a numerical measure of the inequality gap, demonstrating how geometric ideas from calculus can be applied to real-world economic analysis.