how to write the limits for the integration in spherical coordinates to get the volume of
1- the solid inside the cylinder $$r= \sin(\theta)$$ bounded from above by $$z=\sqrt{1-x^2-y^2}$$ and below by the $x$-$y$ plane, in the first quadrant?
2- the cylinder whose base in xy plane lies inside the cardoid $$r=1+ \cos(\theta)$$ and outside the unit circle ... and whose top lies in the plane $z=4$ ...
3- the volume inside the solid $$z= 4-4(x^2+y^2)$$ and above the xy plane .
Note : I know that these integrals are simpler in cylindrical .. But I want to know how to describe using spherical
I have a problem in writing the limits for "rho" in the above problems ..
Also I wonder if we have an horizontal plane bounding the solid from above how to include its equation in the limits..
Can we measure the angle (phy) from positive z- axis if we have an horizontal plane bounding the surface?! I think the surface from above must have a shape like a sphere ..
So can I conclude that if I have shifted paraboloid or shifted cone or a plane bounding the surface from above , it is better to use cylindrical ?