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Thread: Use Cylindrical coordinates to find mass

  1. #1
    VkL is offline
    Oct 2008

    Use Cylindrical coordinates to find mass

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

    How would You set up the integral, I know the density goes in the integrand, but what are the limits of integration?
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  2. #2
    Sep 2009


    Well, I think the limits would be as follows:

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

    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 $\displaystyle \theta$ will go from $\displaystyle 0$ to $\displaystyle 2\pi$

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

    Overall, this the limits should be: $\displaystyle \int{1-x^{2}-y^{2}}^{4}\int_{0}^{1} \int_{0}^{2\pi}d\theta ,dr,dz$
    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.
    Last edited by snaes; Apr 4th 2010 at 08:57 PM. Reason: latex typos
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