Area Between Curves: Integrating With Respect to x

To find the area between two continuous functions on a closed interval [a,b], integrate the vertical distance between the curves with respect to x. The definite integral computes the area using infinitely thin vertical slices, replacing the Riemann sum approximation. This method yields the exact area bounded vertically by the function graphs and horizontally by the interval endpoints.

A Riemann sum estimates area by dividing a region into n vertical strips of equal width and approximating each strip as a rectangle. The height of each rectangle is the vertical distance between the two functions at a selected point. Summing all rectangle areas produces an approximation that improves as the number of strips increases and approaches infinity.

Rectangles with finite width cannot exactly match curved boundaries, creating gaps and overlaps. As the number of strips increases and their width decreases, these approximation errors diminish. In the limiting case where rectangles become infinitely thin, the summation converges to the exact area between the curves.

In economics, the area between the line of perfect equality and the curve representing actual income distribution quantifies income inequality. This geometric measure, called the inequality gap, provides a numerical representation of how income distribution deviates from perfect equality, demonstrating how calculus concepts apply to real-world economic analysis.

Rectangle approximation uses a finite number of vertical strips to estimate area, which provides only an approximation of the true area. Integration with respect to x replaces this summation with a definite integral that computes the exact area using infinitely thin vertical slices, eliminating approximation error.

The functions must be continuous on a closed interval [a,b]. The region between them is bounded vertically by the function graphs and horizontally by the interval endpoints a and b. These conditions ensure the definite integral exists and accurately represents the area.

Integrating with respect to x divides the region into vertical strips and sums their areas. This approach works well when the upper and lower boundaries are easily expressed as functions of x. For regions where horizontal strips are more natural, area between curves integrating with respect to y provides an alternative method.