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The Pappus-Guldinus theorem consists of two parts concerning the volume and surface area of solids of revolution. The first part states that the volume ( V ) of a solid of revolution generated by rotating a plane figure ( A ) about an external axis is given by ( V = A \cdot d ), where ( d ) is the distance traveled by the centroid of ( A ). The second part states that the surface area ( S ) of the solid is given by ( S = P \cdot d ), where ( P ) is the perimeter of the figure and ( d ) is the same distance traveled by the centroid.
Proof Outline: For the volume, consider a plane figure ( A ) with centroid distance ( d ) from the axis of rotation. When ( A ) is rotated, it sweeps out a cylindrical volume, leading to ( V = A \cdot d ) by integrating the circular cross-sections. For the surface area, when the figure is rotated, each infinitesimal segment contributes a cylindrical surface area, leading to ( S = P \cdot d ) through a similar integration process. Both results can be derived using calculus and the properties of centroids and integration.
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