3 (4 points). A solid S consists of the part of a ball of radius 4 that lies
within a cone with its vertex at the origin and consisting of points within
angle phi ³ ¹/6 from the z-axis, as seen from the origin.

  pic1 = ParametricPlot3D[{4 Sin[phi] Cos[theta], \
      4 Sin[phi] Sin[theta], 4 Cos[phi]},{phi,Pi/6, Pi},\
                {theta,0,2 Pi}, PlotRange -> {-4,4}]
               

  pic2 = ParametricPlot3D[{rho Sin[Pi/6] Cos[theta], \
      rho Sin[Pi/6] Sin[theta], rho Cos[Pi/6]},{rho,0,4},\
                {theta,0,2 Pi}, PlotRange -> {-4,4}]
               

  Show[{pic1,pic2}]

In the spherical coordinate system, you would have

  Integrate[rho^2 Sin[phi]^2 Sin[theta]^2  * rho^2 Sin[phi], {phi,Pi/6, Pi} 
  \
        {rho, 0, 4}, 
        {theta, 0,2 Pi}]

The extras after the * are because the differential is rho^2 drho sin(phi)
dtheta dphi. Remember that the order or operations in Mathematica is like
the order of the integral signs (last first). While Mathematica will execute
the command just given, I find it is often better to help her along:

  Integrate[rho^2 Sin[phi]^2 Sin[theta]^2  * rho^2 Sin[phi],{rho,0,4}]

  Integrate[%, {theta,0, 2 Pi}]

  Integrate[%, {phi, Pi/6, Pi}]

  N[%]

Up to Test 3 solutions