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.

In[12]:=

  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}]
               

Out[12]=

  -Graphics3D-

In[13]:=

  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}]
               

Out[13]=

  -Graphics3D-

In[14]:=

  Show[{pic1,pic2}]

Out[14]=

  -Graphics3D-

In the spherical coordinate system, you would have

In[15]:=

  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:

In[16]:=

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

Out[16]=

               3           2
  1024 Sin[phi]  Sin[theta]
  --------------------------
              5

In[17]:=

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

Out[17]=

                  3
  1024 Pi Sin[phi]
  -----------------
          5

In[18]:=

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

Out[18]=

  2048 Pi   384 Sqrt[3] Pi
  ------- + --------------
    15            5

In[19]:=

  N[%]

Out[19]=

  846.831

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