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

Betrachten wir einen Bereich, der durch zwei Kurven begrenzt wird, von denen jede als Funktion von y ausgedrückt wird.Der linke und der rechte Rand, beide Funktionen, sind stetig über das Intervall von y=c bis y=d.Diese horizontalen Linien definieren die vertikale Ausdehnung der Region.Um die Fläche darzustellen, wird die Riemannsche Summe verwendet. Der Bereich ist in dünne horizontale Streifen mit gleicher Höhe Δy und eine Basis unterteilt, die dem horizontalen Abstand zwischen den beiden Randkurven entspricht. Die gesamte geschätzte Fläche wird ermittelt, indem die Flächen aller dünnen Rechtecke summiert werden. Da Δy infinitesimal klein wird, erhält man die Gesamtfläche durch Integration des horizontalen Abstands bezüglich y.Diese Methode wird verwendet, wenn die Randkurven in Bezug auf y beschrieben werden.Betrachten wir eine Dampf-Flüssigkeit-Gleichgewichtsbedingung in einem Gemisch, bei dem die Flüssigkeitsphase x und die Dampfphase y in einem Gleichgewichtszustand koexistieren. Im Diagramm gibt es zwei Kurven – die Diagonalkurve und die Gleichgewichtskurve.Diese umschlossene Fläche wird mit dem Integral der y-Achse berechnet. Das Ergebnis ist die gesamte Separabilitätsfläche, die das Potenzial bietet, die Komponenten über den gesamten Zusammensetzungsbereich zu trennen.

fläche zwischen kurven: integration nach y

Area Between Curves: Integrating With Respect to y

Betrachtet wird ein ebener Bereich, der von zwei Kurven begrenzt wird, die beide als Funktionen der vertikalen Variablen y dargestellt werden. Die linke und rechte Begrenzungskurve sind zwischen y = c und y = d stetig, wobei diese beiden horizontalen Linien die vertikalen Grenzen der Fl&#xE4;che definieren. Da die Grenzen von y und nicht von x abh&#xE4;ngen, eignet sich die Berechnung der Fl&#xE4;che besonders gut mithilfe horizontaler Streifen.
Die Fl&#xE4;che wird mithilfe der Riemannsummenmethode ermittelt. Die Fl&#xE4;che wird in zahlreiche d&#xFC;nne horizontale Streifen gleicher H&#xF6;he &#x394;y unterteilt. Die Breite jedes Streifens entspricht dem horizontalen Abstand zwischen der rechten und der linken Begrenzungskurve bei einem gegebenen y-Wert. Jeder Streifen bildet somit ein d&#xFC;nnes Rechteck, dessen Fl&#xE4;che sich durch Multiplikation seiner H&#xF6;he mit diesem horizontalen Abstand ergibt. Die Summe der Fl&#xE4;chen aller Streifen liefert eine N&#xE4;herung der Gesamtfl&#xE4;che. Mit abnehmender Streifendicke gegen null wird die N&#xE4;herung exakt, und die Gesamtfl&#xE4;che ergibt sich durch Integration des horizontalen Abstands zwischen den Kurven nach y. Diese Vorgehensweise ist besonders vorteilhaft, wenn die Geometrie des Gebiets sich nat&#xFC;rlicher &#xFC;ber y beschreiben l&#xE4;sst.
Diese Methode ist direkt relevant f&#xFC;r die Analyse des Dampf-Fl&#xFC;ssigkeits-Gleichgewichts in Mischungen. In solchen Systemen koexistieren die Zusammensetzungen der Fl&#xFC;ssigphase und der Dampfphase im Gleichgewicht. Ein typisches Gleichgewichtsdiagramm zeigt eine Diagonale, die gleiche Dampf- und Fl&#xFC;ssigkeitszusammensetzungen darstellt, sowie eine Gleichgewichtskurve, die die Phasenbeziehung zwischen beiden Phasen beschreibt. Der Bereich zwischen diesen Kurven repr&#xE4;sentiert die Differenz der Dampf- und Fl&#xFC;ssigkeitszusammensetzung &#xFC;ber den gesamten Zusammensetzungsbereich. Die Fl&#xE4;che dieses Bereichs, berechnet durch horizontale Integration, wird als Gesamttrennfl&#xE4;che bezeichnet. Sie liefert ein quantitatives Ma&#xDF; f&#xFC;r das Trennpotenzial der Gemischkomponenten, wobei gr&#xF6;&#xDF;ere Fl&#xE4;chen auf ein h&#xF6;heres Trennverm&#xF6;gen unter Gleichgewichtsbedingungen hinweisen.

Fläche zwischen Kurven: Integration nach y

Betrachtet wird ein ebener Bereich, der von zwei Kurven begrenzt wird, die beide als Funktionen der vertikalen Variablen y dargestellt werden. Die linke ...

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

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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.

calculus

Horizontal Slices for Area Between Curves

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