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

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 &#x394;y 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 &#x394;y 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.

oppervlakte tussen krommen: integreren met betrekking tot y

Area Between Curves: Integrating With Respect to y

Beschouw een vlak gebied dat wordt begrensd door twee krommen die beide zijn geschreven als functies van de verticale variabele y. De linker- en rechterbegrenzingskrommen zijn continu tussen y = c en y = d, waarbij deze twee horizontale lijnen de verticale grenzen van het gebied bepalen. Aangezien de begrenzingen afhankelijk zijn van y in plaats van x, wordt de oppervlakte het meest geschikt bepaald met behulp van horizontale sneden.
De oppervlakte wordt bepaald met behulp van de Riemannsom-methode. Het gebied wordt opgedeeld in een groot aantal dunne horizontale stroken, elk met een gelijke hoogte, aangeduid als &#x394;y. Voor elke strook wordt de breedte bepaald door de horizontale afstand tussen de rechter- en linkerbegrenzingskromme bij een specifieke waarde van y. Elke strook vormt daarmee een dunne rechthoek, waarvan de oppervlakte wordt verkregen door de hoogte te vermenigvuldigen met deze horizontale afstand. Door de oppervlakten van al deze rechthoeken te sommeren, ontstaat een benadering van de totale oppervlakte van het gebied. Naarmate de dikte van de stroken kleiner wordt en nul nadert, wordt de benadering exact en wordt de totale oppervlakte bepaald door de horizontale afstand tussen de krommen te integreren met betrekking tot y. Deze methode is met name geschikt wanneer de geometrie van het gebied op natuurlijke wijze in termen van y kan worden beschreven.
Deze aanpak is direct relevant voor de analyse van damp-vloeistofevenwichtsgedrag in mengsels. In dergelijke systemen co&#xEB;xisteren de samenstelling van de vloeistoffase en de samenstelling van de dampfase in evenwicht. Een typisch evenwichtsdiagram bevat een diagonale lijn die gelijke damp- en vloeistofsamenstellingen weergeeft, evenals een evenwichtscurve die de faserelatie tussen beide fasen beschrijft. Het gebied dat wordt ingesloten door deze krommen vertegenwoordigt het verschil tussen de damp- en vloeistofsamenstellingen over het volledige samenstellingsbereik. Wanneer dit ingesloten gebied wordt ge&#xEB;valueerd door middel van horizontale integratie, staat de resulterende oppervlakte bekend als de totale scheidbaarheidsoppervlakte. Deze oppervlakte vormt een kwantitatieve maat voor het scheidingspotentieel van de componenten van het mengsel, waarbij grotere oppervlakten wijzen op een groter scheidingsvermogen onder evenwichtsomstandigheden.

Oppervlakte tussen krommen: integreren met betrekking tot y

Beschouw een vlak gebied dat wordt begrensd door twee krommen die beide zijn geschreven als functies van de verticale variabele y. De linker- en ...

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

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JoVE Core

Calculus

Unit

Eenheid

Applications of Integration

Toepassingen van Integratie

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

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

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calculation of volume of solids by integration

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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

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finding volume using cross-sectional area

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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

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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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