# Volume of spherical cap

GROUPS:
 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?
4 years ago
7 Replies
 Luca M 2 Votes 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.
4 years ago
 How would you set it up as an integral?
4 years ago
 Jari Kirma 1 Vote 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.
4 years ago
 The following figure might help...