Lets have a cylinder given by $x^2+y^2=1$ which is cut from the top by plane $z=2$ and bottom by $z=-2$.I am having problem regarding the limits of ρ for the equation ∭ ρ sin^2ϕ dρ dϕ dθ where ϕ is the angle that ρ makes with z axis and $θ$ is the azimuthal angle. I know ρ should start from zero but should it end at(or its upper limit be) cosec(ϕ)???
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You can use the technique in my answer. – Mhenni Benghorbal Dec 21 '13 at 17:56
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you mean i need to split it up?? But here its just a closed cylinder closed by plane surface and not sphere – Smatik Dec 21 '13 at 18:32
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You have different surfaces. You have the plane $z=2$ and the cylinder. – Mhenni Benghorbal Dec 21 '13 at 18:52
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1Oh yes... Got you're point! – Smatik Dec 22 '13 at 05:58
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Here is what you should have
$$ V = 2\left(\int_{0}^{2\pi}\int_{0}^{\tan^{-1}(1/2)}\int_{0}^{2\sec \phi}+\int_{0}^{2\pi}\int_{\tan^{-1}(1/2)}^{\pi/2}\int_{0}^{\csc \phi}\right)\rho^2\sin(\phi)d\rho d\phi d\theta.$$
You should have plots to see what's going on.
Mhenni Benghorbal
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