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March 27, 2017

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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⌋?

  • calculus - ,

    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.

  • calculus - ,

    could you give me a bit help on how to proceed if I were to use a volume integral?

  • calculus - ,

    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.

  • calculus - ,

    ok, thanks

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