4. (8 points). In the cylindrical coordinate system, a spiral ramp can be
defined by the equation
z = theta/3, 0 2 theta 2 4 pi
0 < r 2 1


a) Write the same surface in parametric form as r = x(u,v)i + y(u,v)j + z(u,v)k:

x(u,v) = u Cos[v]

y(u,v) =u Sin[v]

z(u,v) = v/3

where the range for u is 0 to 1 (it is the same as the usual r)

and the range for v is 0 to 4 pi (it is the same as the usual theta).

b) For an explicit integral expression for the surface area of the ramp,
we need to differentiate the position vector by u and by v and take the
cross product

In[16]:=

  rvec[u_,v_] := {u Cos[v], u Sin[v], v/3}

In[17]:=

  D[rvec[u,v], u]
  D[rvec[u,v], v]

Out[17]=

  {Cos[v], Sin[v], 0}

Out[18]=

                          1
  {-(u Sin[v]), u Cos[v], -}
                          3

In[19]:=

  CrossProd = {%%[[2]] %[[3]] - %%[[3]] %[[2]],
                  %%[[3]] %[[1]] - %%[[1]] %[[3]],
                    %%[[1]] %[[2]] - %%[[2]] %[[1]]}

Out[19]=

   Sin[v]  -Cos[v]          2           2
  {------, -------, u Cos[v]  + u Sin[v] }
     3        3

In[20]:=

  Simplify[Sqrt[%[[1]]^2 + %[[2]]^2 + %[[3]]^2 ]]

Out[20]=

       1    2
  Sqrt[- + u ]
       9

In[21]:=

  SurfArea = Integrate[Sqrt[1/9 + u^2], {u,0,1},{v,0, 4 Pi}]

Out[21]=

  2 Pi (3 Sqrt[10] + ArcSinh[3])
  ------------------------------
                9

The value of the integral wass not required, but in case you are interested,

In[22]:=

  N[%]

Out[22]=

  7.89257

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