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Give the spherical coordinates representation of the following solids:

(i) Inside the sphere $ \ x^2+y^2+z^2=4 \ $ and outside the cylinder $ \ x^2+y^2=1 \ $

(ii) Inside the sphere $ \ x^2+y^2+z^2=4 \ $ and above the plane $ \ z=3 \ $

Answer;

(i)

$ 0 \leq \rho \leq 2 \\ 0 \leq \phi \leq \frac{\pi}{4} \\ 0 \leq \theta \leq 2 \pi \ $

Am I right?

Help me out

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Write the equations for the coordinates

\begin{eqnarray} x &=& \rho \sin \theta \cos\phi \\ y &=& \rho \sin \theta \sin\phi \\ z &=& \rho \cos \theta \end{eqnarray}

So that the cylindrical surface can be written as

\begin{eqnarray} x^2+y^2 &=& \rho^2\sin^2\theta\cos^2\phi + \rho^2\sin^2\theta\sin^2\phi \\ &=& \rho^2\sin^2\theta(\cos^2\phi + \sin^2\phi) \\ &=&\rho^2\sin^2\theta \geq 1 \end{eqnarray}

Or equivalentely

$$ \rho |\sin\theta| > 1 $$

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