By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle $x ^ { 2 } + ( y - 5 ) ^ { 2 } = 1$ about the x-axis is

A) $4 \pi ^ { 2 }$
B) $10 \pi ^ { 2 }$
C) $36 \pi ^ { 2 }$
D) $24 \pi ^ { 2 }$
E) $6 \pi ^ { 2 }$

Question

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle $x ^ { 2 } + ( y - 5 ) ^ { 2 } = 1$ about the x-axis is

A) $4 \pi ^ { 2 }$
B) $10 \pi ^ { 2 }$
C) $36 \pi ^ { 2 }$
D) $24 \pi ^ { 2 }$
E) $6 \pi ^ { 2 }$

Quizplus · Accepted Answer

The circle has a radius of 1 and is centered at (0, 5). The centroid of the circle is at (0, 5). By Pappus's Theorem, the volume is the area of the circle ($\pi \times 1^2 = \pi$) times the distance traveled by the centroid, which is $2\pi \times 5 = 10\pi$. Thus, the volume is $\pi \times 10\pi = 10\pi^2$.

By the Pappus Theorem, the Volume of the Solid Formed x2+(y−5)2=1x ^ { 2 } + ( y - 5 ) ^ { 2 } = 1x2+(y−5)2=1

By the Pappus Theorem, the Volume of the Solid Formed $x ^ { 2 } + ( y - 5 ) ^ { 2 } = 1$