# Drawing and computing area of intersection for implicit regions

I want to visualize in 3d intersecting bodies and compute the volume of their intersection at the same time. It would be very nice if I can rotate 3d view and switch between bodies. If it is too hard or takes long time, it is ok for me to have 3 different 3d view in a document.

My bodies are given by inequalities:

1. $x^2+y^2+z^2 \ge 10z;$ $x^2+y^2-2z \le 12$.
2. $x^2+y^2+z^2 -24\le -2z;$ $2 \sqrt{x^2+y^2} \le z+3$.
3. $x^2+y^2+z^2 +6z\le 16;$ $x^2+y^2+z^2 -6z\le 4$.

### Update

My current code:

R =
ImplicitRegion[
x^2 + y^2 + z^2 + 6 z <= 16 && x^2 + y^2 + z^2 - 6 z <= 4, {x, y, z}]
RegionBounds[R]
RegionMeasure[R]
RegionBoundary[R]
Show[{R, DiscretizeRegion[ℛ, RegionBounds[R]]}]


Positive things:

$\quad$I can compute the volume of the intersection.

Negative things:

1. My 3D view show shows something wrong. In my case it should be intersection of two spheres and it shows something close to it, but not what it must be. - SOLVED.
2. I want to extract the equation of the boundaries to compute multiple integrals, but I do not see how to do it. If possible I want to get cylindrical and spherical boundaries as well.
3. I do not see the coordinate axes in my 3D view.

Important question on borders:

With my tiny brain I can deduce borders of the region:

Integrate[1,
{x, -3, 3},
{y, -Sqrt[9 - x^2], Sqrt[9 - x^2]},
{z, 3 - Sqrt[13 - x^2 - y^2], -3 + Sqrt[25 - x^2 - y^2]}]


How can I extract this information from region $R$? I am interested in Cartesian, cylindrical and spherical coordinates at the same time. $R$ must store at least one these representations internally.

• What have you tried so far? Have you converted the formulas to Mathematica syntax? Have you looked at RegionPlot3D, ContourPlot3D, ImplicitRegion, RegionIntersection, Volume? May 21 '17 at 17:14
• I've updated my answer and posted code, please check it. May 21 '17 at 17:30
• Ok, it is getting better, can you please check my question about borders of region? Thanks! May 21 '17 at 17:49
• For the borders: look up CylindricalDecomposition[]. For the axes: Show[(* stuff *), Axes -> True] May 21 '17 at 18:31
• @J. M. Wow, thanks! One last (but very important thing). CylindricalDecomposition gives me Cartesian coordinates. How can I get cylindrical and spherical coordinates from them with Mathematica? Thanks again! May 21 '17 at 18:57 