Use cylindrical coordinates to evaluate the triple integral $\iiint_E (x^2+y^2)dV$, where $E$ is the solid bounded by the circular paraboloid $z=16−4(x^2+y^2)$ and the $xy$-plane.

This is Homework. I dont know what to do. Please help so I can understand. Appreciate it.

  • $\begingroup$ Do you know how to convert from Cartesian to cylindrical coordinates? $\endgroup$ – Américo Tavares Mar 21 '13 at 22:00
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    $\begingroup$ As @AméricoTavares suggests, can you express each of $x$, $y$, and $z$ in terms of $r$, $\theta$, and $z$? $\endgroup$ – Sammy Black Mar 21 '13 at 22:02


  1. You want to express your region in terms of cylindrical coordinates $r, \theta, z$. That will determine the bounds of the integral.
  2. Use $x = r\sin \theta, y = r\cos \theta$ to convert $x^2+y^2$ to $r,\theta,z$ as well.
  3. Use $dV = rdrd\theta dz$ and integrate.

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