Name: disk method for rotated region volume
Uploaded: 2026-01-12T12:24:09.000Z
Duration: 1 min 29 s
Channel: JoVE

Los volúmenes de objetos de formas irregulares pueden determinarse con precisión usando el concepto de sólidos de revolución. Por ejemplo, consideremos una función curva en un plano bidimensional. Girar esta curva alrededor del eje x genera una forma tridimensional simétrica conocida como sólido de revolución. La línea alrededor de la cual gira la región es el eje de la revolución.Cuando la región está directamente contra el eje, produce secciones transversales circulares sólidas. Para determinar el volumen de dicho sólido, se puede utilizar el método del disco.Este método divide el sólido en finas porciones perpendiculares al eje. Cada corte se convierte en un disco con un ancho y radio pequeños iguales a la distancia desde el eje hasta la curva.El volumen de cada disco se obtiene multiplicando su área por su grosor. El volumen total del sólido se aproxima sumando los volúmenes de todos los discos circulares formados.A medida que aumenta el número de discos y su grosor se acerca a cero, la suma pasa a la integral definida del área de la sección transversal en el intervalo dado, dando lugar al volumen exacto.La variable de integración y la expresión para el radio dependen de la orientación del eje.

volúmenes de sólidos de revolución

Volumes of Solids of Revolution

Los vol&#xFA;menes de objetos irregulares pueden determinarse sistem&#xE1;ticamente mediante el concepto de s&#xF3;lidos de revoluci&#xF3;n. Este enfoque parte de una regi&#xF3;n definida por una curva en un plano bidimensional. Al rotar esta regi&#xF3;n sobre una l&#xED;nea fija, conocida como eje de revoluci&#xF3;n, se genera un objeto tridimensional con simetr&#xED;a rotacional. Este tipo de objetos se presenta con frecuencia en aplicaciones de modelado matem&#xE1;tico, f&#xED;sica e ingenier&#xED;a.
Cuando la regi&#xF3;n rotada se encuentra adosada al eje de revoluci&#xF3;n, el s&#xF3;lido resultante consta de secciones transversales circulares sin hueco interior. En este caso, el m&#xE9;todo del disco proporciona una t&#xE9;cnica eficaz para calcular el volumen. El s&#xF3;lido se divide conceptualmente en un gran n&#xFA;mero de cortes muy finos, perpendiculares al eje de rotaci&#xF3;n. Cada corte tiene un peque&#xF1;o grosor y forma un disco cuyo radio est&#xE1; determinado por la distancia desde el eje hasta la curva en esa ubicaci&#xF3;n.
Cada disco representa un s&#xF3;lido geom&#xE9;trico simple cuyo volumen puede calcularse aproximadamente multiplicando su &#xE1;rea circular por su grosor. Aunque un disco individual solo aproxima una parte del s&#xF3;lido, el conjunto de discos aproxima con precisi&#xF3;n el objeto completo. La suma de los vol&#xFA;menes de todos los discos produce una aproximaci&#xF3;n del volumen total. A medida que las secciones se vuelven m&#xE1;s delgadas, esta aproximaci&#xF3;n mejora en precisi&#xF3;n.
En el caso l&#xED;mite, donde el grosor de cada disco se aproxima a cero y el n&#xFA;mero de secciones aumenta considerablemente, la suma de los vol&#xFA;menes de los discos se convierte en un c&#xE1;lculo exacto mediante integraci&#xF3;n definida. La integral acumula las &#xE1;reas de las secciones transversales a lo largo de la longitud del s&#xF3;lido, lo que produce el volumen exacto del s&#xF3;lido de revoluci&#xF3;n. La elecci&#xF3;n de la variable de integraci&#xF3;n, as&#xED; como la expresi&#xF3;n utilizada para representar el radio del disco, depende de si la rotaci&#xF3;n ocurre alrededor de un eje horizontal o vertical.
El m&#xE9;todo del disco es una herramienta fundamental en c&#xE1;lculo y se aplica ampliamente a problemas que involucran formas s&#xF3;lidas suaves generadas por rotaci&#xF3;n. Su simplicidad conceptual e interpretaci&#xF3;n geom&#xE9;trica lo hacen particularmente valioso para analizar sistemas f&#xED;sicos y componentes de ingenier&#xED;a con simetr&#xED;a rotacional.

Volúmenes de Sólidos de Revolución

Los vol&#xFA;menes de objetos irregulares pueden determinarse sistem&#xE1;ticamente mediante el concepto de s&#xF3;lidos de revoluci&#xF3;n. Este enfoque ...

Education

Educación

JoVE Core

Calculus

Cálculo

Unit

Unidad

Applications of Integration

Aplicaciones de la integración

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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 broken into many thin disks. Each disk is taken perpendicular to the axis of rotation. The radius of each disk comes from the distance between the axis and the curve at that point.

Each disk has a small thickness and a circular area. Its volume can be estimated by multiplying that area by the thickness. One disk gives only an approximation, but many disks together give a close model of the whole solid. As the slices become thinner and more numerous, the estimate becomes more accurate.

In the limit, this sum of disk volumes becomes an exact volume through definite integration. The integral adds the cross-sectional areas along the length of the solid. The variable of integration and the radius expression depend on whether the axis of rotation is horizontal or vertical. This method is a key idea in calculus for finding the volume of smooth, rotated shapes.

Disk Method for Rotated Region Volume

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

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

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 the disk method finds the volume of a solid of revolution by slicing a rotated region into thin disks and using definite integration.

Disk Method for Rotated Region Volume | 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

Disk Method for Rotated Region Volume

Description

Transcript

Tags