Calculus triple integral

Question

Find the volume of the solid bounded below by the plane z = 1 and above by the sphere x^2+y^2+z^2=5^2.Write your answer as an exact value in terms of π.

Answer 1

Using Cylindrical Coordinates

The projection of the hemisphere with the plane z = 1 , on the xy- plane is the circle x² + y² = (2√6)²

V = ∫₀^2π ∫_r=0^r=2√6 ∫_z=2^{z=√(r^2-25)} r dz dr dθ =

= ∫₀^2π ∫_r=0^r=2√6 [ rz ] |_z=0^{z=√(r^2-25)} dr dθ =

= ∫₀^2π ∫_r=0^r=2√6r√(r²-25 ) dr dθ. Let w= 25−r² then r dr = − 1/2 dw

= ∫₀^2π ∫_w=25^w=1[ -√w] {-1/2 dw dθ =

=1/2 ∫_θ=0^θ=2π ∫_w=1^w=25w^1/2dw dθ =

= 1/3 ∫_θ=0^θ=2π [w^3/2 ] |₁²⁵ d θ

= 1/3 ∫_θ=0^θ=2π[ 124 ] d θ=

=248π/3