2. (8 points). Find the area of that part of the given surface which lies over the specified region: z = 8 x + 4 y; = is the region between y = x^2 and y = x^6.

Solution. The formula we can use when z is a function of x and y
states that the surface area equals the integral of

In[6]:=

  Clear[f]
  f[x_,y_] := 8 x + 4 y
  FunnyFactor = Sqrt[1 + (D[f[x,y],x])^2 + (D[f[x,y],y])^2]

Out[6]=

In[7]:=

  Area = Integrate[9, {x,0,1}, {y,x^6,x^2}]

Out[7]=

  12
  --
  7