posted by andy on .
Let V be the volume of the region bounded by x^2+y^2≥z^2, z≥0 and x^2+y^2+z^2≤1. What is the value of ⌊100V⌋?
The region is inside a sphere, outside the cone. So, just find the volume of the cone with a spherical cap, and subtract that from half the sphere.
a web search will quickly provide the formulas needed.
Or, you can resort to a volume integral, most easily done using spherical coordinates.
could you give me a bit help on how to proceed if I were to use a volume integral?
the cone is φ = π/4
the sphere is r = 1
dV = r^2 sinφ dφ dθ dr
V = ∫[0,1]∫[0,2π]∫[0,π/4] r^2 sinφ dφ dθ dr
Good luck. Probably ought to verify with cone/sphere formulas to be sure.