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I am trying to solve this problem but I am having difficulties to finish it. I would appreciate of someone can advice me on how to continue

Problem: Calculate $$\iiint_{V} Z\mathrm dV$$ where V is defined by $$ x^2+y^2 \le z^2 $$and$$ x^2+y^2+z^2 \le R^2 with R\gt0$$

Solution Using Cylindrical Coordinates $$\iiint_{V} Z\mathrm dV = \iiint_{V} Z\mathrm rdrdzd\theta $$ $$\iiint_{V} Z\mathrm dV = \iiint_{V} Z\mathrm rdrdzd\theta $$ $$x = rcos\theta, y= rsin\theta $$

Soso
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  • Okay, good, you switched to cylindrical coordinates. That's good. Then you need to think what $x^2 + y^2$ and $R^2$ mean in the cylindrical coordinates. And also reformulate the function $Z$ to be a function of the cylindrical coordinates (if it's possible). Then the problem essentially becomes finding the correct integration limits for $r$, $z$ and $\theta$. – Matti P. Apr 05 '18 at 10:21
  • $r$ and $R$ are different? – user Apr 05 '18 at 10:25
  • @Gimusi: sorry in the problem it is z instead of r. I just corrected it – Soso Apr 05 '18 at 10:27
  • @Soso Please remember that you can choose an answer among the given if the OP is solved, more details here https://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work – user Apr 07 '18 at 20:04

1 Answers1

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HINT

  • make some sketch for the region $E$ in $x-y$,$z-x/y$ plane to uderstand the shape of E
  • define the low $r=f(z)$ and then the set up for the integral in the form

$$\int_0^{2\pi} d\theta \int_0^R dz \int_0^{f(z)} zrdr$$

user
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