The circle x² + y² = a² is rotated around the y-axis to form a solid sphere of radius a. A plane perpendicular to the y-axis at y=a/2 cuts off a spherical cap from the sphere. What fraction of the total volume of the sphere is contained in the cap?
You could find the volume of the cap with the following integral in Mathematica:
\[Pi] Integrate[a^2 - y^2, {y, a/2, a}]
but you could also solve that integral by hand (it’s a very easy one).
The requested ratio can then be calculated as Vcap / Vsphere.
How would you set it up as an integral?
The same result as Luca M gave can also be accomplished leaving more thinking to Mathematica, using Boole, which allows defining region of integration through inequalities:
Integrate[Boole[x^2 + y^2 + z^2 < a^2 && y > a/2],
{y, -Infinity, Infinity}, {x, -Infinity, Infinity}, {z, -Infinity, Infinity},
Assumptions -> a > 0]
Mathematica is not always that clever integrating over implicit regions. Even in this case, it’s best to use order of y, x, z instead of x, y, z to get result quickly.
Luca, this is a very nice plot! Just a general advice - avoid using Subscript or similar if you want to paste readable, much more compact code on Community. Subscript of course will not parse as subscript. Instead of Subscript[r, k] you can always use rk or r if it is a function. For other general tips on smooth posting take a look at: ** How to type up a post: editor tutorial & general tips **.
Thanks “Moderation Team” but how I could post some mathematical formulas (with subscripts and superscripts etc.) mixed with text in the Community threads? I had to convert my text to an image to get the desired result. Is there some other way? Would it be possible to insert something like LaTex code?
For now, if the formulas are not code - images will do. In the future we may implement some functionality for entering formulas.