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Homework Help: Surface Integrals problem

  1. Apr 23, 2007 #1
    1. The problem statement, all variables and given/known data

    Evaluate ∫∫S √(1 + x^2 + y^2) dS where S is the helicoid: r(u, v) = ucos(v)i + usin(v)j + vk, with 0 ≤ u ≤ 4, 0 ≤ v ≤ 3π

    3. The attempt at a solution

    What I tried to do was say x=ucos(v) and y=usin(v), then I plugged those into the sqrt(1+x^2+y^2) eq, which I ended up simplifying to sqrt(1+u) somehow. I then took the cross product of the surface eq differentiated (respect to u) with eq (respect to v). I found the length of that, which I somehow got to be sqrt(2).

    I then said that the intergal was ∫∫sqrt(1+u)*sqrt(2)du*dv
    where the limits of integration are the limits of u and v given above.
    Well this is wrong. Hah!

    Any help would be appreciated.
  2. jcsd
  3. Apr 24, 2007 #2


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    Science Advisor

    I'm not happy with the "somehow"!! That's where you went wrong. The cross product you speak of (also called the 'fundamental vector product') is [itex]sin(v)\vec{i}- cos(v)\vec{j}+ u\vec{k}[/itex]
    (did you forget the "u" multiplying -cos(v) and sin(v) in the derivative with respect to v?) and that has length [itex]\sqrt{1+ u^2}[/itex]. Thus your integral is
    [tex]\int_{u= 0}^4\int_{v=0}^{2\pi}(1+ u^2)dudv[/tex]

    Last edited by a moderator: Apr 24, 2007
  4. Apr 24, 2007 #3
    Thanks for the reply.
    I still cannot seem to get the correct answer. I solved the double integral you provided and was incorrect, then changed the limit on the integral with respect to u to 3pi like the limits given, but still no luck. Any suggestions on what I'm doing wrong? Thanks.
    Last edited: Apr 24, 2007
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