Calculating Volume of a Double-Lobed Cam Using Polar Coordinates

[reddawg]

Homework Statement

The surface of a double lobed cam are modeled by the inequalities:

\frac{1}{4}\leqr\leq\frac{1}{2}(1+cos²θ)

and

-9/(4(x²+y²+9)) ≤ z ≤ 9/(4(x²+y²+9))

Find the volume of the steel in the cam.

Homework Equations

The Attempt at a Solution

I know I need to use a double or triple integral to solve this. I was thinking since I was given r I could change to polar coordinates and solve that way.

Please give me some guidance.

 

[LCKurtz]

Hint: General formula for volumes:
$$\iint_R z_{upper}-z_{lower}~dA$$

 

[reddawg]

Ok, that makes sense. Wasn't sure if I could do that.

R would be the r given. Theta is from 0 to 2*pi.

Therefore I can convert the bounds to polar coordinates.

My z_upper - z_lower is taken from the z given in the inequality.

x^2 + y^2 become r^2 and I can integrate completely from there.

Right?

 

[LCKurtz]

If you mean what I think you mean, yes.

 

[reddawg]

Ha ha ok. Thanks LCKurtz.

When I evaluate the integral (using a calculator) I get 0.79993.

This seems awfully low to be a volume of an shape like this.

- I checked it twice for errors, I think it's accurate.

 

[LCKurtz]

That looks like it might be about right. Here's a picture (I had a little time to waste):

 

[reddawg]

Wow, thanks for wasting your time for me! ;-)

I guess the cam is pretty small so the volume is more reasonable than I thought.