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 ˛ š/6 from the z-axis, as seen from the origin.

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 ˛ š/6 from the z-axis, as seen from the origin.

a) sketch this region


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


  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 -> {0,4}]


  Show[{pic1,pic2}]

In the cylindrical coordinate system, you would have


  Integrate[r^2 Cos[theta]^2  * r, {theta,0, 2 Pi} \
        {r, 0, 4 Sin[Pi/6]},
        {z, r Cot[Pi/6],Sqrt[16-r^2]}]

The extra r is because the differential is r dr dtheta dz. 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[r^2 Cos[theta]^2  * r,{z, r Cot[Pi/6],Sqrt[16-r^2]}]


  Integrate[%, {r, 0, 4 Sin[Pi/6]}]


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


  N[%]

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