9.7:
Theorems of Pappus and Guldinus
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 x-axis.
Consider a differential line element. When revolved, it generates a ring, which is integrated to obtain the entire surface area.
The first theorem states that the area of a surface of revolution is the product of the generating curve'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.
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.
9.7:
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° rotation, their respective formulae would need to be altered accordingly.
Önerilen Okuma
- Hibbeler, R.C. (2016). Engineering Mechanics: Statics. Fourteenth Edition, New Jersey: Pearson. Section 9.3, Pp. 502-503.
- Meriam, J.L., Kraige, L.G. and Bolton, J.N. (2016). Engineering Mechanics: Statics and Dynamics. Eighth Edition, Singapore: John Wiley & Sons. Section 5.5 Pp. 261-262.