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Happiness
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Pappus's centroid theorems were discovered 17 centuries ago, when calculus wasn't invented yet. How are these theorems proved without using calculus?
"The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by its geometric centroid."
The centroid of an object is its center of mass supposing its density is uniform.
"The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by its geometric centroid."
Quotes from https://en.wikipedia.org/wiki/Pappus's_centroid_theorem
"The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by its geometric centroid."
The centroid of an object is its center of mass supposing its density is uniform.
"The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by its geometric centroid."
Quotes from https://en.wikipedia.org/wiki/Pappus's_centroid_theorem