Area Between Curves: Problem Solving

A region can be enclosed by three curves: a square root function, a reflected cube root function, and a linear function. The linear function intersects each of the other two curves, and these intersection points determine where the boundary of the enclosed region changes. Because different curves serve as the upper and lower boundaries in different parts of the graph, the area cannot be found using a single setup over the entire interval.

To compute the area, the region is first divided into two segments at the intersection point where the “top” and “bottom” boundary curves switch. Over the interval from x = 0 to x = 1, the square root curve lies above the reflected cube root curve, so the area in this segment is obtained by integrating the vertical distance between these two curves with respect to x, then evaluating that definite integral across the interval. Over the interval from x = 1 to x = 4, the linear function becomes one of the active boundaries, and a different pair of curves defines the upper and lower edges of the region. The same procedure is applied: integrate the vertical distance between the appropriate upper and lower boundary curves and evaluate the definite integral on this second interval.

The total enclosed area is then found by adding the two results. Splitting the region at the boundary-change point ensures that each integral represents a single, consistent “upper minus lower” relationship, which is necessary for an accurate geometric interpretation.

This piecewise area method is widely used in welfare economics to quantify total economic surplus. In a standard market diagram, the demand curve and the supply curve form the boundaries of the surplus region, and the equilibrium point determines where the relevant interval ends. The total surplus is calculated by integrating the vertical difference between demand and supply from zero to the equilibrium quantity. The resulting area represents the net economic benefit generated by market exchange, combining consumer and producer gains in a single measure.