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A cylindrical drill with radius 1 is used to bore a hole throught the center of a sphere of radius 6.?
A cylindrical drill with radius 1 is used to bore a hole throught the center of a sphere of radius 6. Find the volume of the ring shaped solid that remains.
1 Answer
- kbLv 77 years agoFavorite Answer
Using cylindrical coordinates:
x^2 + y^2 + z^2 = 6^2 ==> z = ±√(36 - r^2).
Projecting the solid onto the xy-plane gives a region bounded between r = 1 and r = 6.
So, the volume ∫∫∫ 1 dV equals
∫(θ = 0 to 2π) ∫(r = 1 to 6) ∫(z = -√(36 - r^2) to √(36 - r^2)) 1 * (r dz dr dθ)
= 2π ∫(r = 1 to 6) 2r√(36 - r^2) dr
= 2π * (-2/3)(36 - r^2)^(3/2) {for r = 1 to 6}
= (4π/3) 35^(3/2).
I hope this helps!
