- #1
TheSpaceGuy
- 25
- 0
Homework Statement
Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 =4, above the xy-plane, and below the cone z=sqrt(x^2 + y^2).
The Attempt at a Solution
Use Cylindrical Coordinates.
Note that r ≤ z ≤ √(4 - r^2).
These sphere and cone intersect when x^2 + y^2 + (x^2 + y^2) = 4
==> x^2 + y^2 = 2, a circle with radius √2.
So, the projection onto the xy-plane is a disk centered at the origin with radius √2.
Thus, the volume equals
∫∫∫ 1 dV
= ∫(t = 0 to 2π) ∫(r = 0 to √2) ∫(z = r to √(4 - r^2)) 1 (r dz dr dt)
= 2π ∫(r = 0 to √2) r (√(4 - r^2) - r) dr
= π ∫(r = 0 to √2) (2r √(4 - r^2) - 2r^2) dr
= π [(-2/3)(4 - r^2)^(3/2) - 2r^3/3] {for r = 0 to √2}
= (-2π/3) [r^3 + (4 - r^2)^(3/2)] {for r = 0 to √2}
= (-2π/3) [(2√2 + 2√2) - (0 + 8)]
= (π/3) (16 - 8√2).
Is this correct?