Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of and about the y-axis.
A) 144$\pi$
B) 108$\pi$
C) 648$\pi$
D) 864$\pi$

Question

Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of  and  about the y-axis.
A) 144$\pi$ 
B) 108$\pi$ 
C) 648$\pi$ 
D) 864$\pi$

Quizplus · Accepted Answer

The region bounded by the given graphs is a rectangle with length $4\pi$ and width $11-(-11)=22$. Thus, the area of this region is $88\pi$. 
To apply the theorem of Pappus, we need to find the distance between the axis of rotation (in this case, the y-axis) and the centroid of the region. The centroid of a rectangle lies at the midpoint of its two diagonals, so the centroid of our region is at $(0,0)$. Therefore, the distance between the y-axis and the centroid is simply the distance from the y-axis to the origin, which is 0. 
Now, we can apply the theorem of Pappus, which tells us that the volume of the solid obtained by revolving the given region about the y-axis is equal to the product of the area of the region and the distance traveled by the centroid. Since the distance traveled by the centroid is 0, the volume of the solid is also 0. 
Using the choices given, we see that only option C gives 0 as the volume of the solid, so the correct answer is C.

Use the Theorem of Pappus to Find the Volume of the Solid