I need to find the volume of this paraboloid, $x^2+y^2=z,z=4y,$ using triple integrals, but have trouble determining the limits of integration.
I know it's an inverted paraboloid with the plane $z=4y$ cutting at top.
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If you intersect the surfaces $$x^2+y^2=z=4y$$ you get the circle $x^2+(y-2)^2=4$ and you see that in the following plot. Now, take a look at that circle to find the ranges for $$\theta,~~\text{and}~~~r$$ In fact, I want to use Cylindrical Coordinates here. As you see, the volume is symmetric with respect to $x$ and $y$ so we consider the part which is established on $x\ge 0, y\ge 0$. The second Fig below shows that $$r|_{0}^{4\sin\theta}, ~\theta|_{0}^{\pi/2}$$ So we have $$V=2\int_{\theta}\int_{r}\int_{z=r^2}^{4r\sin\theta}rdzdrd\theta$$
Mikasa
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+1 Your skill in drawing figures is surprising for me. :) Happy Eid! (kam peida boodin ye moddat!) – Mahdi Khosravi Aug 08 '13 at 13:31
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Thanks! ishalla ebadate shoma ham ghabool bashe. – Mahdi Khosravi Aug 08 '13 at 13:39
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@BabakS. Thanks !!, can you also tell me what will the limits be in cartesian co-ordinates ? as i am not able to see that yet and it would help me understand it better. – Aman Mittal Aug 08 '13 at 15:09
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@AmanMittal: $$2\int_{y=0}^4\int_{x=0}^{2+\sqrt{4-(y-2)^2}}\int_{z=x^2+y^2}^{4y}dxdydz$$ – Mikasa Aug 08 '13 at 15:16
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@BabakS. I was almost there :) , Thanks you so much – Aman Mittal Aug 08 '13 at 18:26
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@BabakS. i generally have problem with solving this kind of complex surfaces. Would you suggest some read or technique to work around them ? – Aman Mittal Aug 08 '13 at 18:33
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@AmanMittal: If I wanted to solve this problem in my class, I would suggest students to do it by Cylindrical coordinates. It make the integrals easier. Adam's book is a very good help. – Mikasa Aug 08 '13 at 19:11
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Nice work, dear friend! – amWhy Aug 09 '13 at 00:07
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in the cylindrical co-ordinates, i get the limit for $\theta$ from $0$ to $\pi$ and $r$ from $0$ to $4sin\theta$ . does that look good to you ? – Aman Mittal Sep 24 '13 at 19:01