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

Pappus and Guldinus developed theorems to find the surface area and volume of any body of revolution.
To generate the surface area, revolve a plane curve of known length around the&#160; x-axis.
Consider a differential line element. When revolved, it generates a ring, which is integrated to obtain the entire surface area.&#160;
The first theorem states that the area of a surface of revolution is the product of the generating curve&#39;s length and the distance the centroid of the curve travels while generating the surface.
Similarly, to find the volume of revolution, a plane area is revolved around a non-intersecting axis.
Consider a differential area, generating a volume element when revolved. The differential volume is then integrated to determine the entire volume.&#160;
The second theorem states that the volume of a body of revolution is the product of the generating area and the distance the centroid of the area travels while generating the volume.
In both cases, the formula changes accordingly if the revolution is only through an angle.

theorems of pappus and guldinus

The two theorems developed by Pappus and Guldinus are widely used in mathematics, engineering, and physics to find the surface area and volume of any body of revolution. This is done by revolving a plane curve around an axis that does not intersect the curve to find its surface area or revolving a plane area around a non-intersecting axis to calculate its volume.
For finding the surface area, consider a differential line element that generates a ring with surface area dA when revolved. Integrating this differential area determines the surface area of the revolution, which is described by the first theorem. The first theorem states that the surface area of revolution is equal to the product of the length of the generating curve and the distance traveled by its centroid during the generation of the surface area.
Similarly, for calculating the volume of revolution, consider a differential area element generating a ring with differential volume when revolved. Integrating this differential volume determines the volume of revolution, as per the second theorem. The second theorem of Pappus and Guldinus states that the volume of revolution is equal to the product of the generating area and the distance traveled by its centroid in generating such volume. Understandably, if either curve or area is revolved through an angle instead of a complete 360&#176; rotation, their respective formulae would need to be altered accordingly.

Theorems of Pappus and Guldinus

The two theorems developed by Pappus and Guldinus are widely used in mathematics, engineering, and physics to find the surface area and volume of any body ...

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Center of Gravity and Centroid

KEY TERMS AND CONCEPTS

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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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...

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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.

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Surface Area and Volume by Centroid Paths

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