I'm trying to write using spherical coordinates the set
$$W=\{(x,y,z)\in\mathbb{R}\mid x^{2}+y^{2}+z^{2}\geq1, x^{2}+y^{2}+(z-2)^{2}\leq4, z\geq\sqrt{x^{2}+y^{2}}\}$$
I could draw the solid and I saw two different parts: the one in the top, from the cone to the bigger sphere and the one in the bottom, among the spheres and out of the cone.
To the top part, I got
$$0\leq\rho\leq2,$$$$ 0\leq\theta\leq2\pi$$
$$2\leq z\leq\sqrt{4-r^{2}}+2$$
using cylindrical coordinates.
But my problem is with the bottom part. I don't know how to see the limits to $\rho$; and I think $0\leq\phi\leq\pi/2$ and $0\leq\theta\leq2\pi$.
Best Answer
The required region is constituted of $3$ parts.
My notation: $\theta$ = Polar angle, where angle is taken with respect to the vector joining the two spheres, pointing two origin, $\phi$ = Azimuthal angle.
Parametrization of upper spherical cap: $$ \begin{pmatrix} &2 \sin(\theta) \cos(\phi) \\ & 2 \sin(\theta) \sin(\phi)\\ &2 \cos(\theta) \end{pmatrix}$$ with limits $ \theta \in \bigg ( \dfrac{\pi}{2} ,\dfrac{3 \pi}{2} \bigg) $,$ \phi \in (0, 2 \pi )$
Parametrization of lower spherical cap: $$ \begin{pmatrix} &G \sin(\theta) \cos(\phi) \\ & G \sin(\theta) \sin(\phi)\\ &G \cos(\theta) \end{pmatrix}$$ Where $$G = 2 \cos \theta - \sqrt{4 \cos^2 \theta -3}$$
with limits: $ \theta \in \bigg ( 0 , \arccos \bigg( \cfrac{ 2 - 2^{-1/2}}{\sqrt{5 - 2 \sqrt{2} } } \bigg) $,$ \phi \in (0, 2 \pi )$
Parametrization of conical section: $$ \begin{pmatrix} &H \sin(\theta) \cos(\phi) \\ & H \sin(\theta) \sin(\phi)\\ &H \cos(\theta) \end{pmatrix}$$
Where $$H = \cfrac{2}{\cos \theta + \sin \theta} $$ with limits: $ \theta \in \bigg ( \arccos \bigg( \cfrac{ 2 - 2^{-1/2}}{\sqrt{5 - 2 \sqrt{2} } }\bigg) , \dfrac{ \pi}{2} \bigg) $,$ \phi \in (0, 2 \pi )$