Center of Mass (Triple Integral)

[Reefy]

Homework Statement

T is the solid bounded by the cylinder y^2+z^2=4 and the planes x=0 and x=3. The mass density at a point P of T is directly proportional to the distance between P and the yz-plane.

Find the center of mass of the solid T.

Homework Equations

y^2+z^2=4

x=0

x=3

The Attempt at a Solution

I drew the solid and got a cylinder extending from x=0 (yz-plane) all the way to x=3 with a radius of 2.

I also attempted to set up an integral but I think my main problem is figuring out what the density to integrate is.

I set up my integral as the integral from x=0 to x=3, the integral from y= -2 to y=2, and the integral from
z= -√(4-y^2) to z=√(4-y^2) dz dy dx.

Is that correct? I don't know how to go about determining my p(x,y,x) aka my density.Edit: Initially I tried to use x as my density but I couldn't integrate that so I tried y and then z but none of them worked out.

 

[Reefy]

No one knows?

 

[tiny-tim]

Hi Reefy!

(try using the X² button just above the Reply box )

I set up my integral as the integral from x=0 to x=3, the integral from y= -2 to y=2, and the integral from
z= -√(4-y^2) to z=√(4-y^2) dz dy dx.

Is that correct?

Yes

(but in this case the density is constant over each circular slice, so you could just use πr², and integrate only over x )

I don't know how to go about determining my p(x,y,x) aka my density.

Edit: Initially I tried to use x as my density but I couldn't integrate that …

You can't integrate ∫ x dx ?

(or ∫∫∫ x dxdydz)