Name: piecewise area between curves
Uploaded: 2026-06-30T08:00:37.538Z
Duration: 1 min 28 s
Channel: JoVE

Consideremos una región delimitada por una función raíz cuadrada, una función de raíz cúbica reflejada y una función lineal.La función lineal intersecta las otras dos curvas, formando los límites de la región.Para encontrar el área, la región se divide en dos segmentos basados en el cambio en las curvas de frontera en los puntos de intersección.En el intervalo de x = 0 a x = 1, el área se determina integrando la distancia vertical entre las curvas de frontera superior e inferior respecto a x, y evaluando la integral definida sobre este intervalo.En el intervalo de x = 1 a x = 4, un par diferente de curvas de frontera define la parte superior e inferior de la región. El área sobre este segmento también se determina integrando la distancia vertical correspondiente y evaluando la integral definida.El área total encerrada es la suma de las dos integrales definidas.Este método se utiliza en la economía del bienestar para medir el superávit económico total. Las curvas de oferta y demanda forman los límites de la región. El superávit total se calcula integrando la diferencia vertical entre las curvas desde cero hasta la cantidad de equilibrio, representando el beneficio económico neto.

Área entre curvas: resolución de problemas

Area Between Curves: Problem Solving

Una regi&#xF3;n puede estar encerrada por tres curvas: una funci&#xF3;n de ra&#xED;z cuadrada, una funci&#xF3;n de ra&#xED;z c&#xFA;bica reflejada y una funci&#xF3;n lineal. La funci&#xF3;n lineal interseca cada una de las otras dos curvas, y estos puntos de intersecci&#xF3;n determinan d&#xF3;nde cambia el l&#xED;mite de la regi&#xF3;n encerrada. Dado que diferentes curvas sirven como l&#xED;mites superior e inferior en distintas partes del gr&#xE1;fico, el &#xE1;rea no se puede calcular con un solo planteamiento en todo el intervalo.
Para calcular el &#xE1;rea, la regi&#xF3;n se divide primero en dos segmentos en el punto de intersecci&#xF3;n donde se intercambian las curvas que act&#xFA;an como l&#xED;mites superior e inferior. En el intervalo de x = 0 a x = 1, la curva de ra&#xED;z cuadrada se encuentra por encima de la curva de ra&#xED;z c&#xFA;bica reflejada; por lo tanto, el &#xE1;rea en este segmento se obtiene integrando la distancia vertical entre estas dos curvas con respecto a x y evaluando la integral definida en todo el intervalo. En el intervalo de x = 1 a x = 4, la funci&#xF3;n lineal se convierte en uno de los l&#xED;mites activos, y un par distinto de curvas determina los l&#xED;mites superior e inferior apropiados de la regi&#xF3;n. Se aplica el mismo procedimiento: integrar la distancia vertical entre las curvas que act&#xFA;an como l&#xED;mites superior e inferior apropiados y evaluar la integral definida en este segundo intervalo.
El &#xE1;rea total encerrada se obtiene sumando ambos resultados. Dividir la regi&#xF3;n en el punto de cambio de l&#xED;mite garantiza que cada integral represente una relaci&#xF3;n &#xFA;nica y coherente de &#xAB;l&#xED;mite superior menos l&#xED;mite inferior&#xBB;, necesaria para una interpretaci&#xF3;n geom&#xE9;trica precisa.
Este m&#xE9;todo de &#xE1;rea por partes se utiliza ampliamente en la econom&#xED;a del bienestar para cuantificar el excedente econ&#xF3;mico total. En un diagrama de mercado est&#xE1;ndar, la curva de demanda y la curva de oferta forman los l&#xED;mites de la regi&#xF3;n excedente, y el punto de equilibrio determina d&#xF3;nde termina el intervalo relevante. El excedente total se calcula integrando la diferencia vertical entre la demanda y la oferta desde cero hasta la cantidad de equilibrio. El &#xE1;rea resultante representa el beneficio econ&#xF3;mico neto generado por el intercambio de mercado, combinando las ganancias del consumidor y del productor en una &#xFA;nica medida.

Una regi&#xF3;n puede estar encerrada por tres curvas: una funci&#xF3;n de ra&#xED;z cuadrada, una funci&#xF3;n de ra&#xED;z c&#xFA;bica reflejada y una ...

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Education

Educación

JoVE Core

Calculus

Cálculo

Unit

Unidad

Applications of Integration

Aplicaciones de la integración

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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 the point where the upper and lower boundaries switch. From x = 0 to x = 1, the square root curve lies above the reflected cube root curve. The area for that part is found by integrating the vertical distance between those two curves with respect to x.

From x = 1 to x = 4, the linear function becomes part of the boundary. A different pair of curves then defines the upper and lower edges of the region. The same method is used again. The definite integral of the vertical distance is evaluated on the second interval, and the two results are added to get the total enclosed area.

Splitting the region at the boundary-change point keeps each integral in a simple upper-minus-lower form. That makes the geometric meaning of the area clear and accurate. This piecewise method also appears in welfare economics, where it helps measure total economic surplus. In a standard market diagram, the demand curve and supply curve form the bounds of the surplus region, and the equilibrium point marks the end of the interval. The total surplus is found by integrating the vertical difference between demand and supply from zero to the equilibrium quantity. The area represents the net economic benefit of market exchange and combines consumer and producer gains into one measure.

Piecewise 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-integrating-with-respect-to-y

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...

area between curves: problem solving

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 split an area between curves into intervals, use upper minus lower integrals, and connect the method to total economic surplus in markets.

Piecewise 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

Piecewise Area Between Curves

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