Problem 43 - Calculus of a Single Variable

Question

State the Theorem of Pappus.

Step-by-Step Solution

Answer

The Theorem of Pappus states that the volume of a geometric solid obtained by revolving a planar region about an axis is equal to the product of the area of the region and the path traversed by the centroid of the area during the revolution.

Step 1: Understand the Theorem of Pappus

The Theorem of Pappus applies to geometric solids obtained by revolving a plane region about an axis. Given the area \(A\) of the region and the distance \(d\) that the centroid of the area travels during the revolution, the theorem states that the volume \(V\) of the solid formed is \(V = A \times d\). This theorem also extends to the surface area of the solid, using the length of a curve instead of area.

Step 2: Explaining relevant terminology

The centroid of a shape is its geometric center, a single point that could be considered as the 'average' of all the points in the shape. In the context of the theorem, it is the point around which the shape revolves to produce the solid. Revolution in this context refers to the movement of a shape about an axis, producing a solid figure.