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The solid region E that is the right circular cone of height $10$ and diameter $5$, sitting with its base on the $xy$-plane centered at $(0,0,0)$.

Use reasoning to compute the following triple integral without converting it into iterated integrals.

$$\iiint_E x+ydV$$

I am not sure how to begin this problem without using iterated integrals? I graphed the cone using Mathemtica as well as the graph of $z=x+y$.

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  • $\begingroup$ Have you studied mass, center of mass and etc.? And I think the cone's base's center should be $\;(0,0,0)\;$ , as you're working in the space $\;\Bbb R^3\;$ . $\endgroup$ – DonAntonio Apr 13 '16 at 21:23
  • $\begingroup$ yes i have but this does not seem to be a mass related problem $\endgroup$ – Bennett Lerud Apr 13 '16 at 21:28
  • $\begingroup$ Well, Joriki's answer already solved...and without using any physics. $\endgroup$ – DonAntonio Apr 13 '16 at 21:33
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The cone is symmetric with respect to inversion of any axis lying in the $xy$-plane, in particular the $x$ and $y$ axis. Thus the integral is zero by symmetry.

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  • $\begingroup$ what do you mean by inversion of any axis in the xy-plane? it makes sense to me that the integral would be zero, however, i am not sure I understand how to explain it $\endgroup$ – Bennett Lerud Apr 14 '16 at 16:35
  • $\begingroup$ @BennettLerud: Sorry, I forgot to answer this question. I mean that if you choose any axis in the $xy$ plane, and in particular the $x$ axis or the $y$ axis, and invert the coordinate along that axis (i.e. apply a reflection in the plane through the origin that's perpendicular to that axis), the cone remains invariant. Hence the integral over $x$ is equal to the integral over $-x$, and hence is zero; and likewise for $y$. $\endgroup$ – joriki Apr 20 '16 at 18:18

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