The volume of a spherical cap by integrating and using Cartesian coordinates

[nejla]

Dear all,

How can I derive the volume of a spherical cap by integration and using the Cartesian coordinate system.

The sphere is located at the (0,0,0) coordinates and its radius is set to r. The height of the cap is also set to (r-h).

I googled a lot but I couldn't find it.

I would be enormously grateful if you could help me.

 

[Eynstone]

Integrate z(x,y)=sqrt(r^2 -x^2 -y^2) on the circle x^2 +y^2 = r^2- h^2. The substitution x=R cost, y=Rsint would simplify the integration.

 

[nejla]

Dear Eynstone,

Thank you sooooo much. You are generous. No DOUBT.
I asked this question because I wanted to solve another problem and now it seems that it is more complicated than what I was expecting.

May I ask you about that question?

How can I find the volume of the intersection of two spherical caps where Cap1 and Cap2 are the result of the intersection of the sphere, S, and planes P1 and P2 , respectively. the angle between P1 and p2 is Theta (0<Theta<180). S is placed at the (0,0,0) coordinates and the radius of the sphere is set to r. The height of Cap1 and Cap2 are set to h1 and h2, respectively. P1 is (z=h1) but P2 is (ax+by+cz=d) [we know both plane equations].

I was thinking that I can calculate the volume as:

f1= Integral(1,z,-sqrt(r^2-x^2-y^2),sqrt(r^2-x^2-y^2))
f2=Integral(f1,y,-sqrt(r^2-h1^2-y^2),-sqrt(r^2-h1^2-y^2))
f3=Integral(f2,x, x_min, x_max)

I can find x_min and x_max

volume of the intersection of Cap1 and Cap2 = f3

Now, I am quite sure that it is wrong.
Do you have any idea that how can I calculate this volume as you are a real genius mathematician?

Many many thanks again,
Nejla