Problem 74 - Essential Calculus

Question

State the Theorem of Pappus.

Step-by-Step Solution

Answer

The Theorem of Pappus describes a relation between areas for plane geometry and a relation between volumes for solid geometry. The plane version states that if two plane figures share a common centroid, the area of the figure is equal to the product of the areas of the figures plus twice the product of the areas of figures formed by half-lines drawn from the centroid. The solid version states that the volume of a solid of revolution generated by revolving a plane figure is the product of the area of the figure and the distance travelled by its geometric centroid.

Step 1: Understand the context

To answer this, we should understand the context. The Theorem of Pappus is two theorems in geometry, attributed to Pappus of Alexandria. One is regarding plane geometry and the other one is about solid geometry.

Step 2: State Theorem of Pappus for plane geometry

The plane version, also known as Pappus's Centroid Theorem deals with areas. Its statement is: If A and B are two plane figures that share a common centroid, the area of the figure is equal to the product of the areas of A and B plus twice the product of the areas of figures formed by half-lines drawn from the centroid to points on A and B.

Step 3: State Theorem of Pappus for solid geometry

The solid version deals with volumes. Its statement is: The volume of a solid of revolution generated by the revolution of a plane figure about an external axis is equal to the product of the area A of the figure and the distance d travled by its geometric centroid. More formally, it's expressed as: \(V = A \cdot d\).