Area Between Curves: Integrating With Respect to y

Integrate with respect to y when boundary curves are naturally expressed as functions of y rather than x. This approach uses horizontal slices instead of vertical ones, making calculations simpler when the region's left and right boundaries depend on the vertical variable. Horizontal integration is especially useful for regions where x-values are difficult to express as single functions of y.

The Riemann sum divides the region into thin horizontal strips of equal height Δy. Each strip's area equals its height times the horizontal distance between the left and right boundary curves. Summing all rectangle areas approximates the total area. As Δy approaches zero, this approximation becomes exact, yielding the definite integral.

The integral represents the horizontal distance between the two boundary curves, integrated over the vertical interval from y = c to y = d. This distance function, when integrated with respect to y, gives the exact area of the region bounded by the two continuous curves. The result is a precise measurement of the enclosed planar region.

The total separability area is found by integrating the horizontal distance between the equilibrium curve and the diagonal line with respect to composition. This enclosed region quantifies the difference between vapor and liquid phase compositions across the entire composition range, providing a measure of separation potential for mixture components.

The equilibrium curve captures the phase relationship between liquid and vapor compositions in a mixture at equilibrium. Combined with the diagonal line representing equal compositions, the area between these curves indicates the separation capability. Larger enclosed areas represent greater potential for component separation under equilibrium conditions.

Horizontal lines at y = c and y = d establish the lower and upper bounds of the integration interval. These lines define where the region begins and ends vertically, ensuring the integral captures the complete area between the boundary curves. Without these limits, the integration would lack defined endpoints.

Horizontal integration uses horizontal slices and integrates with respect to y, while vertical integration uses vertical slices and integrates with respect to x. Choose horizontal integration when curves are functions of y; choose vertical when they're functions of x. The method selected depends on which variable makes the boundary expressions simpler and more natural.