Finding the domain of integration in spherical coordinate of a shifted cylinder

[gsingh2011]

So I've done some problems where a sphere intersects with a cylinder and I needed to find the volume of the intersected region using triple integrals. For example, if I needed to find the domain of integration for the intersection of the sphere $$x^2+y^2+z^2=a^2$$ and the cylinder $$x^2+y^2=\frac{a^2}{4}$$ then it would be $$D=\{(\rho, \phi, \theta)|0\le \rho \le a, 0 \le \phi \le \frac{\pi}{6}, 0 \le \theta \le 2\pi\}$$

Now let's say that the cylinders equation was something like $$(x-\frac{a}{2})^2+y^2=\frac{a}{2}$$ Now the cylinder is shifted by $$\frac{a}{2}$$, so how do I find the ranges for $$\phi$$ and $$\theta$$?

Side note: How do I do inline LaTeX?

 

[fresh_42]

Inline formulas are within double ## tags.
Here is a list of coordinate transformations:
https://en.wikipedia.org/wiki/List_of_common_coordinate_transformationsYou have to define $\phi$ and $\theta$ in terms of $x,y,z$.