# In terms of spherical coordinates describe the following solids

In terms of spherical coordinates describe the following solids:

(i) Inside the sphere $\ x^2+y^2+z^2=16 \$ and above the plane $\ z=3 \$

(ii) Between the spheres $\ x^2+y^2+z^2=5 \$ and $x^2+y^2+z^2=3 \$

(i)

The spherical coordinate is

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

The description is given below:

$\frac{3}{\cos \phi} \leq \rho \leq 4 \\ 0 \leq \phi \leq \pi \\ 0 \leq \theta \leq 2 \pi \$

(ii)

I can not describe the same ranges for the second case.

Help me

HINT

For the first check the bound for $\phi$

$\frac{3}{\cos \phi} \leq \rho \leq 4 \\ \color{red}{0 \leq \phi \leq {pi}}\\ 0 \leq \theta \leq 2 \pi \$

For the second simply

$\sqrt 3 \leq \rho \leq \sqrt 5 \\ 0 \leq \phi \leq \pi \\ 0 \leq \theta \leq 2 \pi \$

• what should be the bound for $\ \phi \$ for first part? I think $\ 0 \leq \phi \leq \pi \$ Is not it ? – M. A. SARKAR Mar 16 '18 at 14:54
• @yourmath the plane x-z (y=0) then the intersection between the line $z=3$ and $x^2+z^2=16$ we find $x=\pm \sqrt 7$ then $\pi_{max}=\arctan \frac{\sqrt 7}{3}$ – gimusi Mar 16 '18 at 14:57

$\sqrt{3} \leq \rho \leq \sqrt{5}$
$0 \leq \phi \leq \pi$
$0 \leq \theta \leq 2\pi$

--- rk