Pappus's Centroid Theorem states that the surface area and volume of a solid of revolution can be calculated using the centroid of the shape being revolved. Specifically, when a plane figure is revolved around an external axis, the surface area generated is equal to the product of the length of the figure and the distance traveled by its centroid, while the volume generated is the product of the area of the figure and the distance traveled by its centroid. This theorem connects geometric properties with calculus concepts, providing a powerful tool for analyzing complex shapes.
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