I need help to verify if my working is correct in order to get the volume of a given solid bounded by some surfaces.

Given a solid $S$ in $\mathbb{R}^3$ defined by :

\begin{equation*} x^2 \ + \ y^2 \ \le8y \ , \ x^2 \ + \ y^2 \ge 4z~\text{and}~z\ge0. \end{equation*}

I used cylindrical coordinates after drawing the domain. The solid is given by a region bounded by an elliptic paraboloid and a cylinder of radius $4$ with $z\ge 0$ . I bounded ($ \ r, \phi , z)$ by bounding $0\le\phi\le\pi$, $0\le r\le8\sin(\phi)$ and $0\le z\le {r^2\over4}$ And integrated to finally obtain $\int_0^{\pi}\sin^4(\phi)d\phi$= $3\pi\over8$. Is my work correct??

  • $\begingroup$ one problem with your solution is that it is $r^2$ rather than $r$ that is bounded by $8y$. $\endgroup$
    – WW1
    May 9, 2015 at 21:18
  • $\begingroup$ Yes sorry thats a duplicate i didnt notice. $r^2$ $\le 8y$ $\rightarrow$ $r^2$ $\le 8rsin(\phi)$ $\rightarrow$ $r$ $\le 8sin(\phi)$. $\endgroup$
    – mandez
    May 9, 2015 at 21:30
  • $\begingroup$ Related post on meta: This question is an exact duplicate of: Page Not Found $\endgroup$ Jun 15, 2016 at 12:38


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