5.2
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…
Consider a region bounded by two curves, each expressed as a function of y.
The left and the right boundary, both functions, are continuous over the interval from y=c to y=d.
These horizontal lines define the vertical extent of the region.
To find the area, the Riemann sum method is used. The region is partitioned into thin horizontal strips of equal height Δy and a base equal to the horizontal distance between the two boundary curves. The total estimated area is found by summing the areas of all thin rectangles. As Δy becomes infinitesimally small, the total area is obtained by integrating the horizontal distance with respect to y.
This method is used when the boundary curves are described in terms of y.
Consider a vapor-liquid equilibrium plot, where x and y represent the liquid- and vapor-phase mole fractions. The diagonal line, y = x, is a reference line representing equal liquid and vapor mole fractions.
The enclosed area is calculated by integrating the horizontal distance between the two curves with respect to y, providing a measure of the relative ease of separation across the full composition range.
View the full transcript and gain access to JoVE Core videos
Q1: When should you integrate with respect to y instead of x?
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.
Q2: How does the Riemann sum method work for horizontal slices?
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.
Q3: What does the integral represent when integrating with respect to y?
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.
Q4: How is the total separability area calculated in vapor-liquid equilibrium?
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.
Q5: What role does the equilibrium curve play in phase separation analysis?
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.
Q6: Why are horizontal lines important in defining the region's vertical extent?
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.
Q7: How does horizontal integration compare to vertical integration for area calculations?
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.