MHB Vector Calculus - Cylindrical Co-ordinates

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To find the volume of the ellipsoid defined by the equation ${R}^{2}+{3z}^{2}=1$ using cylindrical coordinates, the Jacobian is indeed $r$, leading to the integral $\iiint (r)dzdrd\theta$. The key challenge lies in determining the appropriate bounds for $z$, $r$, and $\theta$. For this ellipsoid, $z$ can be expressed in terms of $r$, and the limits for $r$ will depend on the projection of the ellipsoid onto the $xy$-plane. The integration process involves setting up the correct limits based on the geometry of the ellipsoid and the cylindrical coordinate system.
RigB
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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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