Lets have two spheres, the middle point of sphere 1 is on the edge of sphere2(see picture). If I want to calculate the volume that is inside this region of the two spheres, do I need to use cylindrical coordinates or spherical? And can someone show me how to find the boundaries
The volume that is shared by the two spheres is a volume of revolution which could be found by a single integral.
Note that the equation of the right hand side sphere is $$(x-1)^2+y^2=4$$
The section of that sphere which is in the second and the third quadrant is $$\int _{-1}^0 \pi y^2=\int _{-1}^0 \pi [4-(x-1)^2]dx =5\pi /3$$
Thus the total volume is twice that which is $$ V=\frac {10\pi}{3}$$