Vector Calculus - Cylindrical Co-ordinates

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SUMMARY

The discussion focuses on using cylindrical coordinates to calculate the volume of the ellipsoid defined by the equation ${R}^{2}+{3z}^{2}=1$. The integral setup involves the Jacobian $r$, leading to the expression $$\iiint (r)dzdrd\theta$$. Key challenges include determining the appropriate bounds for the variables $z$, $r$, and $\theta$, as well as clarifying the function to be integrated. The ellipsoid's geometry requires careful consideration of these limits to perform the volume calculation accurately.

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  • Understanding of cylindrical coordinates in multivariable calculus
  • Familiarity with Jacobians and their role in changing variables for integrals
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  • Experience with triple integrals and volume calculations
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I have a question, to use cylindrical coordinates to find the volume of the ellipsoid ${R}^{2}+{3z}^{2}=1$.

I know for cylindrical coordinates the Jacobian is $r$ so I have some integral:

$$\iiint (r)dzdrd\theta$$

However I am struggling to work out the bounds of the integral for $z,r,\theta$ and also what I am integrating. Please may someone explain the method for working out the limits in this example and what I am integrating? I should be okay to do the integral itself from then on.

I would really appreciate it. Thank you.
 
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I think there's some information missing here. What you have so far is an ellipse in $R$ and $z$.
 

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