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

Consideremos una región delimitada por dos curvas, cada una expresada como función de y.El límite izquierdo y derecho, ambas funciones, son continuos en el intervalo de y=c a y=d.Estas líneas horizontales definen la extensión vertical de la región.Para mostrar el área, se utiliza el método de suma de Riemann. La región se divide en franjas horizontales delgadas de igual altura Δy y una base igual a la distancia horizontal entre las dos curvas de contorno. El área total estimada se obtiene sumando las áreas de todos los rectángulos delgados. A medida que Δy se vuelve infinitesimalmente pequeño, el área total se obtiene integrando la distancia horizontal respecto a y.Este método se utiliza cuando las curvas de frontera se describen en términos de y.Consideremos una condición de equilibrio vapor-líquido en una mezcla donde la fase líquida, x, y la fase de vapor, y, coexisten en un estado de equilibrio. En el gráfico, hay dos curvas: la diagonal y la curva de equilibrio.Esta área encerrada se calcula usando la integral del eje y. El resultado es el área total de separabilidad, lo que permite que los componentes se separen a lo largo de todo el rango de composición.

Área entre curvas: integración con respecto a y

Area Between Curves: Integrating With Respect to y

Considere una regi&#xF3;n plana delimitada por dos curvas, ambas expresadas en funci&#xF3;n de la variable vertical, y. Las curvas l&#xED;mite izquierda y derecha son continuas entre y = c e y = d, y estas dos l&#xED;neas horizontales definen los l&#xED;mites verticales de la regi&#xF3;n. Dado que los l&#xED;mites dependen de y en lugar de x, la forma m&#xE1;s adecuada de calcular el &#xE1;rea es mediante cortes horizontales.
El &#xE1;rea se obtiene mediante el m&#xE9;todo de las sumas de Riemann. La regi&#xF3;n se divide en varias franjas horizontales finas, cada una con el mismo espesor, denotado por &#x394;y. Para cada franja, el ancho viene dado por la distancia horizontal entre la curva derecha y la curva izquierda en un valor particular de y. Por lo tanto, cada franja forma un rect&#xE1;ngulo fino, y su &#xE1;rea se calcula multiplicando su espesor por esta distancia horizontal. La suma de las &#xE1;reas de todos estos rect&#xE1;ngulos produce una aproximaci&#xF3;n del &#xE1;rea total de la regi&#xF3;n. A medida que el espesor de las franjas disminuye y se acerca a cero, la aproximaci&#xF3;n se vuelve exacta y el &#xE1;rea total se determina integrando la distancia horizontal entre las curvas respecto a y. Este enfoque es especialmente &#xFA;til cuando la geometr&#xED;a de la regi&#xF3;n se describe de forma m&#xE1;s natural en t&#xE9;rminos de y.
Este m&#xE9;todo tiene relevancia directa en el an&#xE1;lisis del comportamiento del equilibrio vapor&#x2013;l&#xED;quido en mezclas. En estos sistemas, las composiciones de la fase l&#xED;quida y de la fase de vapor coexisten en equilibrio. Un diagrama de equilibrio t&#xED;pico incluye una l&#xED;nea diagonal que representa composiciones id&#xE9;nticas de vapor y l&#xED;quido, as&#xED; como una curva de equilibrio que captura la relaci&#xF3;n de fases entre ambas. La regi&#xF3;n cerrada entre estas curvas representa la diferencia entre las composiciones de vapor y l&#xED;quido en todo el rango de composici&#xF3;n. Cuando esta regi&#xF3;n se calcula mediante integraci&#xF3;n horizontal, el &#xE1;rea resultante se conoce como &#xE1;rea total de separabilidad. Un &#xE1;rea mayor indica una mayor capacidad de separaci&#xF3;n en condiciones de equilibrio.

Área entre curvas: Integración con respecto a y

Considere una regi&#xF3;n plana delimitada por dos curvas, ambas expresadas en funci&#xF3;n de la variable vertical, y. Las curvas l&#xED;mite izquierda 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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