# Volume of a double-lobed cam

1. Mar 30, 2014

### reddawg

1. The problem statement, all variables and given/known data
The surface of a double lobed cam are modeled by the inequalities:

$\frac{1}{4}$$\leq$r$\leq$$\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.

2. Relevant equations

3. 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.

2. Mar 30, 2014

### LCKurtz

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

3. Mar 30, 2014

### 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?

4. Mar 30, 2014

### LCKurtz

If you mean what I think you mean, yes.

5. Mar 30, 2014

### 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.

6. Mar 30, 2014

### LCKurtz

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

#### Attached Files:

• ###### cam.jpg
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7. Mar 30, 2014

### 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.