Name: horizontal slices for area between curves
Uploaded: 2026-07-24T11:47:53.303Z
Duration: 1 min 30 s
Channel: JoVE

考虑一个由两条曲线围成的区域，每条曲线均表示为关于 y 的函数。左边界和右边界均为函数，且在 y=c 到 y=d 的区间上连续。这些水平线定义了该区域的垂直范围。为了求面积，采用黎曼和方法。将区域分割成高度相等、为 Δy 的细长水平条带，其底边长度等于两条边界曲线之间的水平距离。通过累加所有细长三角形的面积，得到总面积的估计值。当 Δy 趋于无穷小时，通过对水平距离关于 y 进行积分，即可得到总面积。当边界曲线用 y 表示时，使用此方法。考虑一个气液平衡相图，其中 x 和 y 分别表示液相和气相的摩尔分数。y = x 的对角线是一条参考线，代表液相与气相摩尔分数相等的情况。通过在 y 方向上对两条曲线之间的水平距离进行积分来计算封闭区域的面积，从而提供在整个组成范围内分离难易程度的度量。

曲线间面积：关于 y 的积分

Area Between Curves: Integrating With Respect to y

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area-between-curves-integrating-with-respect-to-y

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Applications of Integration

积分的应用

KEY TERMS AND CONCEPTS

关键术语与概念

horizontal slices for area between curves

Area between curves can be found with horizontal slices when both boundaries are written as functions of y. The left and right curves are continuous from y = c to y = d, and those lines set the vertical limits of the region. Because the curves depend on y, the area is best measured by moving across the shape with thin horizontal strips.

The method uses a Riemann sum. The region is divided into many thin strips with the same height, delta y. For each strip, the width is the horizontal distance between the right boundary curve and the left boundary curve at a chosen value of y. Each strip is a thin rectangle, so its area is height times width.

Adding the areas of all the rectangles gives an approximation of the total area. As the strips become thinner and approach zero, the approximation becomes exact. The area is then found by integrating the horizontal distance between the curves with respect to y.

This approach is useful when a region is easier to describe with y than with x. It also connects to vapor-liquid equilibrium in mixtures. In that setting, the liquid-phase composition and the vapor-phase composition exist together in equilibrium.

A typical equilibrium diagram includes a diagonal line where vapor and liquid compositions are equal. It also includes an equilibrium curve that shows the phase relationship between the two phases. The region between these curves shows the difference between vapor and liquid compositions across the full composition range.

When this enclosed region is measured using horizontal integration, the result is called the total separability area. This area gives a way to measure how well the components of a mixture may be separated under equilibrium conditions. A larger area suggests greater separation capability.

Horizontal Slices for Area Between Curves

horizontal-slices-for-area-between-curves

area between curves: integrating with respect to x

area-between-curves-integrating-with-respect-to-x

Vertical Slices for Area Between Curves

Area between curves can be found by using vertical slices and a definite integral. For two continuous functions on a closed interval from a to b, the region is bounded by their graphs on the sides and by the interval endpoints across the bottom and top edges of the setup. A first estimate comes from splitting the interval into many narrow vertical strips of equal width. Each strip is treated like a rectangle. Its height is the vertical difference between the two functions at a chosen point...

area between curves: integrating with respect to y

Area between curves can be found with horizontal slices when both boundaries are written as functions of y. The left and right curves are continuous from y = c to y = d, and those lines set the vertical limits of the region. Because the curves depend on y, the area is best measured by moving across the shape with thin horizontal strips. The method uses a Riemann sum. The region is divided into many thin strips with the same height, delta y. For each strip, the width is the horizontal distance...

area between curves: problem solving

Piecewise Area Between Curves

Piecewise area between curves uses more than one integral to find the size of a region on a graph. In this example, the region is enclosed by a square root function, a reflected cube root function, and a linear function. The line intersects each of the other two curves, and those intersection points show where the boundary of the region changes. The area cannot be found with one setup over the full interval because the top and bottom curves are not the same everywhere. The region is split at...

calculation of volume of solids by integration

calculation-of-volume-of-solids-by-integration

Slicing Methods for Solid Volume

Slicing methods help find the volume of solids with changing cross-sections. A basic solid, like a rectangular box, is easy to measure because its base area stays the same along its height. Many real objects do not have that regular shape, so elementary geometry is not enough. The slicing method solves this by breaking a solid into many very thin slices. These slices are taken perpendicular to a chosen axis, often the x-axis. Each slice has a small thickness, written as delta x, and a...

volumes of solids of revolution

Disk Method for Rotated Region Volume

The disk method is used to find the volume of a solid formed by rotating a region around an axis. The region is first drawn in a two-dimensional plane. When that region spins around a fixed line, called the axis of revolution, it creates a three-dimensional shape with rotational symmetry. These shapes appear often in math models, physics, and engineering. When the region touches the axis of revolution, the solid has no hollow space inside. Its cross-sections are circular, so the shape can be...

finding volume using cross-sectional area

finding-volume-using-cross-sectional-area

Tetrahedron Volume from Triangle Slices

Tetrahedron volume can be found by adding the areas of triangular slices. This method uses cross-sectional area, which is the area of a flat slice taken through a 3D solid. It is useful when a solid changes shape in a regular way and a simple geometric formula is not available. A tetrahedron is a good example. It has a height h and an equilateral triangle base with side length a. The coordinate system places the origin at the top vertex, and the y-axis goes downward along the height to the...

problem solving: volume

Finding the Volume of a Jet Fuel Tank

The volume of a jet fuel tank can be found with solids of revolution. The tank is modeled by rotating a two-dimensional region around the x-axis. This creates a three-dimensional shape that is symmetric about the axis of rotation. The region runs from 0 to 2 meters, so the tank’s shape can be measured along that length. Because the boundary curve touches the axis, the disk method works well here. The disk method breaks the solid into many very thin circular slices that are perpendicular to the...

average value of a function

Estimating Function Behavior with Average Value

The average value of a function gives one number that summarizes how a quantity changes over a closed interval. It can also be viewed as the height of a rectangle whose area matches the net area under the curve. That net area includes both positive and negative parts of the function, so the result reflects the function’s overall behavior. A greenhouse temperature over a 24-hour period is a practical example of this idea. The temperature may rise and fall throughout the day, but it is still...

key terms and concepts

Learn how to find area between curves using horizontal slices, Riemann sums, and integration with respect to y, with a vapor-liquid equilibrium example.

Horizontal Slices for Area Between Curves | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

Horizontal Slices for Area Between Curves

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