Use Pappus's Theorem to find the volume of the torus obtained when the region inside the circle x^2 + y^2 = a^2 is revolved about the line x = 2a.
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Use Pappus's Theorem to find the volume of the torus obtained
when the region inside the circle $\displaystyle x^2 + y^2 = a^2$ is revolved about the line $\displaystyle x = 2a$
Pappus' Theorem . . . The volume of a solid of revolution is equal to:
the area of the region times the distance travled by its center of mass.
The area of the circle is $\displaystyle \pi a^2$. .The center of mass is its center.
The distance travelled is the circumference of a circle with radius $\displaystyle 2a\!:\;\;2\pi(2a) = 4\pi a$
Therefore, the volume is: .$\displaystyle V \;= \;(\pi a^2)(4\pi a) \;= \;4\pi^2a^3$