1. ## Volumes by slicing

I have been attempting to do the question with different methods.
I can do this question by the shells method, but I want to know how to do it by volumes by slicing/washer method.

Question:
A circle of radius 2 is given by $\displaystyle x^2+y^2 = 4$ and is revolved about the line $\displaystyle x=5$. Using the washer/volume by slicing method, prove that the resulting volume (a TORUS) is $\displaystyle 40 {\pi}^2$.

Any help would be greatly appreciated!

2. This is known as the Theorem of Pappus.

If the radius is b and the distance from the line to the center of the circle is a, then:

By symmetry, the centroid of a circular region is its center. Thus, the distance traveled by the centroid is $\displaystyle 2{\pi}a$.

Since the area of a circle is $\displaystyle {\pi}r^{2}$, we get

$\displaystyle V=(2{\pi}a)({\pi}b^{2})=2{\pi}(5)(4{\pi})=40{\pi}^ {2}$

EDIT: I'm sorry, you said the washer method. The method I posted, is that the one you know?. I would assume. It resembles shells and is most commonly use for these types of problems.

When you're revolving a circle in a torus fashion, the idea is to find the area of the circle and multiply it by the circumference of the circle created when it revolves.

You could do it this way. This is using washers:

$\displaystyle (10{\pi})(2)\int_{-2}^{2}\sqrt{4-x^{2}}dx=40{\pi}^{2}$