Name: surface area and volume by centroid paths
Uploaded: 2023-09-22T11:49:32.000Z
Duration: 1 min 10 s
Channel: JoVE

Pappus y Guldinus desarrollaron teoremas para encontrar el área superficial y el volumen de cualquier cuerpo de revolución.
Para generar el área de superficie, gire una curva plana de longitud conocida alrededor del eje x.
Considere un elemento de línea diferencial. Al girar, genera un anillo, que se integra para obtener toda la superficie. 
El primer teorema establece que el área de una superficie de revolución es el producto de la longitud de la curva generadora y la distancia que recorre el centroide de la curva mientras genera la superficie.
De manera similar, para encontrar el volumen de revolución, un área plana gira alrededor de un eje que no se intersecta.
Considere un área diferencial, generando un elemento de volumen cuando se gira. A continuación, se integra el volumen diferencial para determinar el volumen completo. 
El segundo teorema establece que el volumen de un cuerpo de revolución es el producto del área generadora y la distancia que recorre el centroide del área mientras genera el volumen.
En ambos casos, la fórmula cambia en consecuencia si la revolución es solo a través de un ángulo.

<ol>
	<li>Hibbeler, R.C. (2016). Ingeniería Mecánica: Estática. Decimocuarta edición, Nueva Jersey: Pearson. Sección 9.3, págs. 502 y 503.</li>
	<li>Meriam, J.L., Kraige, L.G. y Bolton, J.N. (2016). Ingeniería Mecánica: Estática y Dinámica. Octava edición, Singapur: John Wiley & Sons. Sección 5.5, págs. 261 y 262.</li>
</ol>

los teoremas de pappus y guldinus

Los dos teoremas desarrollados por Pappus y Guldinus son ampliamente utilizados en matemáticas, ingeniería y física para encontrar el área de superficie y el volumen de cualquier cuerpo de revolución. Esto se hace al girar una curva plana alrededor de un eje que no intersecta la curva para encontrar su área de superficie, o al girar un área plana alrededor de un eje no intersecto para calcular su volumen. Para encontrar el área de superficie, consideremos un elemento de línea diferencial que genera un anillo con área de superficie dA al girar. Integrar esta área diferencial determina el área de superficie de la revolución, que está descrita por el primer teorema. El primer teorema establece que el área de superficie de la revolución es igual al producto de la longitud de la curva generadora y la distancia recorrida por su centroide durante la generación del área de superficie. De manera similar, para calcular el volumen de revolución, consideremos un elemento de área diferencial que genera un anillo con volumen diferencial al girar. Integrar este volumen diferencial determina el volumen de revolución, según el segundo teorema. El segundo teorema de Pappus y Guldinus establece que el volumen de revolución es igual al producto del área generadora y la distancia recorrida por su centroide al generar dicho volumen. Comprensiblemente, si tanto la curva como el área se giran a través de un ángulo en lugar de una rotación completa de 360°, sus fórmulas respectivas deberán ser modificadas en consecuencia.

Los teoremas de Pappus y Guldinus

Los dos teoremas desarrollados por Pappus y Guldinus son ampliamente utilizados en matemáticas, ingeniería y física para encontrar el área de superficie y el volumen de cualquier cuerpo ...

Education

Educación

JoVE Core

Mechanical Engineering

Ingeniería Mecánica

Unit 1

Unidad 1

Center of Gravity and Centroid

Centro de gravedad y centroide

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

Theorems of Pappus and Guldinus

surface area and volume by centroid paths

Pappus and Guldinus theorems help find the surface area and volume of bodies of revolution. A body of revolution is formed when a plane curve or plane area is turned around an axis that does not intersect it.

For surface area, a curve is revolved around a non-intersecting axis. A small line element on the curve creates a thin ring as it moves. Adding these ring areas gives the total surface area of the solid.

The first theorem says that this surface area equals the length of the generating curve multiplied by the distance traveled by its centroid. The centroid is the geometric center of the curve as it moves.

For volume, a plane area is revolved around a non-intersecting axis. A small area element creates a ring with a tiny volume as it turns. Integrating these small volumes gives the total volume of revolution.

The second theorem says that this volume equals the generating area multiplied by the distance traveled by its centroid. If the curve or area turns through an angle instead of a full 360° rotation, the formula must be adjusted. These ideas are useful in mathematics, engineering, and physics.

Surface Area and Volume by Centroid Paths

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key terms and concepts

Learn how Pappus and Guldinus theorems use centroid motion to find surface area and volume of bodies of revolution, with 360° and angled rotations.

Surface Area and Volume by Centroid Paths | JoVE K12

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Surface Area and Volume by Centroid Paths

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