Mass of a Cardioid: Find Mass & Center of Mass

[Locoism]

Homework Statement

Find the mass and center of mass of the region bounded by the cardioid r =
1 + cos(θ), assuming the density function is given by ρ = r.

The Attempt at a Solution

The first part is simple enough, I set up the integral

\int_{0}^{2\pi}\int_{0}^{1+cos(\theta)}r dr d(\theta)

I find the mass is 3pi/2

Now for the centre of, how do I set up the integrals? I know it would be the double integral of xρ drdθ, and then yρ drdθ, but do I keep the same limits? x would be rcos(θ) and y is rsin(θ), but this gives me some really weird functions to integrate.

 

[Dick]

I'm not sure what's so weird about the functions to integrate, but you are making a mistake already. In polar coordinates to find area you integrate the volume element dxdy=r*dr*d(theta). So I think you just computed the area. If the density is also r, that should give you an extra factor of r.

 

[Locoism]

Oh so in that case I get 5/6 pi. But still, what about the centre of mass? I need My and Mx, do I keep the same limits of integration?

 

[Dick]

Oh so in that case I get 5/6 pi. But still, what about the centre of mass? I need My and Mx, do I keep the same limits of integration?

Well, I don't get 5*pi/6. I'd check that. But, yes, keep the same limits of integration. Why not? Just put a factor of x or y into the integrand and work them out. Then you divide by total mass, right?

 

[Locoism]

sorry 5pi/3. Thank you!