Set up in spherical coordinates the volume of the solid betw

Set up in spherical coordinates, the volume of the solid between x^2 + y^2 + z^2 = 1 and x^2 + y^2 + z^2 = 9 where y lessthanorequalto 0, z greaterthanorequalto 0.

Solution

put z = r*cos(theta) theta = [-90,90]

x = r*sin(theta)*cos(phi)

y = r*sin(theta)*sin(phi) when theta = [-90,0], phi = [0,180]

when theta = [0,90], phi = [180,360]

so solids are , in 1st r =1

=> sin²(theta)*cos²(phi) + sin²(theta)*sin²(phi) + cos²(theta) = 1/r²

in 2nd r = 3

=> sin²(theta)*cos²(phi) + sin²(theta)*sin²(phi) + cos²(theta) = 9/r\'²