# Integral to find the volume of a torus

togame

## Homework Statement

Find the volume of the torus formed when the circle of radius 2 centered at (3,0) is revolved about the y-axis. Use geometry to evaluate the integral.

## Homework Equations

Formula for the semi-circle: $y=\sqrt{4-x^2}$
Solving for x give $x=\pm\sqrt{4-y^2}$

## The Attempt at a Solution

I was thinking of solving the integral from -2 to 2 using both the positive and negative sides, then evaluate the integral and then multiply by 2, but I'm not so sure about my equation.

$$2\int_{-2}^2\pi\big( (3-\sqrt{4-y^2})^2 - (3+\sqrt{4-y^2})^2 \big) \mathrm d y$$

I just seem lost on this one :(

janhaa

## Homework Statement

Find the volume of the torus formed when the circle of radius 2 centered at (3,0) is revolved about the y-axis. Use geometry to evaluate the integral.

## Homework Equations

Formula for the semi-circle: $y=\sqrt{4-x^2}$
Solving for x give $x=\pm\sqrt{4-y^2}$

## The Attempt at a Solution

I was thinking of solving the integral from -2 to 2 using both the positive and negative sides, then evaluate the integral and then multiply by 2, but I'm not so sure about my equation.
$$2\int_{-2}^2\pi\big( (3-\sqrt{4-y^2})^2 - (3+\sqrt{4-y^2})^2 \big) \mathrm d y$$
I just seem lost on this one :(
I believe the equation is:

$$V=\int_{-2}^2\pi\big( (3+\sqrt{4-y^2})^2 - (3-\sqrt{4-y^2})^2 \big) \mathrm d y$$

$$V=\int_{-2}^2\pi\cdot 12(\sqrt{4-y^2}) \mathrm d y$$

then trigonometric substitution...

Homework Helper

## Homework Statement

Find the volume of the torus formed when the circle of radius 2 centered at (3,0) is revolved about the y-axis. Use geometry to evaluate the integral.

## Homework Equations

Formula for the semi-circle: $y=\sqrt{4-x^2}$
Solving for x give $x=\pm\sqrt{4-y^2}$

## The Attempt at a Solution

I was thinking of solving the integral from -2 to 2 using both the positive and negative sides, then evaluate the integral and then multiply by 2, but I'm not so sure about my equation.

$$2\int_{-2}^2\pi\big( (3-\sqrt{4-y^2})^2 - (3+\sqrt{4-y^2})^2 \big) \mathrm d y$$

I just seem lost on this one :(

Edit : Got beat to it.

Science Advisor
Homework Helper
the way to do this is to use pappus' theorem. the product of the distance traveled by the center of the circle by the area of the circle.