spherical triple integral to find the volume

Question

Le‎t G₃ be the s‎ol‎id in the f‎irst oct‎ant bound‎ed ab‎‎ove by the sph‎ere x² + y² + (z − 2)² = 4 and bel‎ow by the c‎o‎ne z = √(x² + y²). U‎se a spher‎ical tr‎iple integr‎al to fin‎‎d the vol‎ume of G₃.

Answer 1

right is that the transformation to spherical

transforms dxdydz to r²sin(a)drdθda,,,,,,where a is angle measured from z axis to line formed connecting origin to point. Note a will only go from pi/4 to 0. θ will go from 0 to pi/2, and r from 0 to 2.

V=∫∫∫r²drdθsinada,

=∫∫(√2/2)r²drdθ,,,,,,integral of sinada = -cosa

=∫(pi/2)(√2/2)r²dr

=(8/3)(pi/2)(√2/2)

=(2pi√2)/3