Volume/Pappus' Theorem -- 30 mins remaning

**thedoge** · December 7th 2006, 09:34 PM

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.

**Soroban** · December 8th 2006, 03:41 AM

Hello, thedoge!

Do you know Pappus' Theorem? .Did you make a sketch?

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 $x = 2a$

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 $\pi a^2$ . .The center of mass is its center.

The distance travelled is the circumference of a circle with radius $2a\!:\;\;2\pi(2a) = 4\pi a$

Therefore, the volume is: . $V \;= \;(\pi a^2)(4\pi a) \;= \;4\pi^2a^3$

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