Area Between Curves: Integrating With Respect to y

Consider a planar region bounded by two curves that are both written as functions of the vertical variable, y. The left and right boundary curves are continuous between y = c and y = d, and these two horizontal lines define the vertical limits of the region. Because the boundaries depend on y rather than x, the area is most appropriately evaluated using horizontal slices.

The area is obtained using the Riemann sum method. The region is divided into many thin horizontal strips, each having an equal height, denoted as delta y. For each strip, the width is given by the horizontal distance between the right boundary curve and the left boundary curve at a particular value of y. Each strip, therefore, forms a thin rectangle, and its area is found by multiplying its height by this horizontal distance. Adding the areas of all such rectangles produces an approximation of the total area of the region. As the thickness of the strips becomes smaller and approaches zero, the approximation becomes exact, and the total area is determined by integrating the horizontal distance between the curves with respect to y. This approach is especially useful when the geometry of the region is more naturally described in terms of y.

This method has direct relevance in the analysis of vapor–liquid equilibrium behavior in mixtures. In such systems, the liquid-phase composition and the vapor-phase composition coexist in equilibrium. A typical equilibrium diagram includes a diagonal line representing equal vapor and liquid compositions, along with an equilibrium curve that captures the phase relationship between the two phases. The region enclosed between these curves represents the difference between the vapor and liquid compositions across the entire composition range. When this enclosed region is evaluated using horizontal integration, the resulting area is known as the total separability area. This area provides a quantitative measure of the potential for separating the components of the mixture, with larger areas indicating greater separation capability under equilibrium conditions.