# Thread: Use Cylindrical coordinates to find mass

1. ## Use Cylindrical coordinates to find mass

Use Cylindrical coordinates to find the mass of the solid region bound above by the plane $z=4$, below by the paraboloid $z=1-x^2-y^2$ and on the side by the cylinder $x^2+y^2 = 1$ if the density is given by $\rho(x,y,z)=k *sqrt{[(x^2+y^2)]}$

How would You set up the integral, I know the density goes in the integrand, but what are the limits of integration?

2. ## limits

Well, I think the limits would be as follows:

z limit it bound by the plane and paraboloid.
$\int_{1-x^{2}-y^{2}}^4$

For the cylindrical coordinates: You need to go the entire way around the circel(if you look at the paraboloid from top down in a 2D perspective it looks like a circle), so $\theta$ will go from $0$ to $2\pi$

Next, what is the radius?...by looking at the cylinder equation its 1.
$\int_{0}^{1} \int_{0}^{2\pi}d\theta ,dr$

Overall, this the limits should be: $\int{1-x^{2}-y^{2}}^{4}\int_{0}^{1} \int_{0}^{2\pi}d\theta ,dr,dz$
Just remember to change to cylindrical coordinates after you have completed the first integral which is in cartesian.

Lastly, for checking your work, its pretty easy to use, I have an example written to get you started.
http://www.wolframalpha.com/input/?i...dx+from+0+to+1