How do I find the integration boundaries for the volume integral of $$M:=\{(x,y,z)\in\mathbb{R}^3:x^2+cy^2-f(z)^2\le 0\}$$ $f:[a,b]\rightarrow\mathbb{R}^+_0$, $c> 0$.
Volume integral boundaries
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calculus
integration
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0I think you may have an infinite surface there. For example, with $\;c=1\;,\;\;f(z)=z\;$ , we get $$\;x^2+y^2-z^2\le0\iff z^2\ge x^2+y^2$$ and this is all the space *above and below* the double cone$\; z^2=x^2+y^2\;$ , which has no fixed, finite boundaries...Perhaps some info is lacking? – 2017-01-28
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0That's all the information I got. Is the surface not bounded due to $z\in[a,b]$? – 2017-01-28
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0@V Yes, I missed that piece of info...but we must then assume $\;f\;$ is continuous. Let me check now... – 2017-01-28
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0@DonAntonio Does something like $f(a)^2\le f(z)^2\le f(b)^2$, $-f(z)\le cy^2\le f(z)$ and $-\sqrt{f(z)^2-cy^2}\le x\le\sqrt{f(z)^2-cy^2}$ maybe look ok? – 2017-01-28
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0$-f(z)\le \sqrt{c}y\le f(z)$ of course – 2017-01-28