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Parameterised surfaces

  1. Jun 10, 2009 #1
    1. The problem statement, all variables and given/known data
    Use parametrization to express the area of the surface as a double integral. tilted plane inside cylinder, the portion of the plane y+2z=2 inside the cylinder x^2+y^2=1



    2. Relevant equations
    the area of a smooth surface
    r(u,v)=f(u,v)i+g(u,v)j+h(u,v)k a<=u<=b c<=v<=d
    is
    A=integral from c to d ( integral from a to b(|r subu X r subv|))dudv


    3. The attempt at a solution
    x=x z=z y=2-2z
    r(x,z)=xi+(2-2z)j+zk
    r subx=i
    r subz=-2j+k
    r subx X r subz=| i j k |=-j-2k
    | 1 0 0 |
    | 0 -2 1 |

    |r subx X r subz |=sqrt(5)

    Area=integral 0 to 2pi(integral from 0 to 1(sqrt(5)r))drd(theta)
    =sqrt(5)*pi


    yet the answers have:
    Area=integral 0 to 2pi(integral from 0 to 1(sqrt(5)r/2))drd(theta)
    =sqrt(5)*pi/2


    can someone please help?
     
  2. jcsd
  3. Jun 10, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    You are integrating over the region in the x-y plane right? So you want r(x,y). And you want to integrate |r subx X r suby|. Express the surface in terms of x and y coordinates. Not x and z.
     
  4. Jun 10, 2009 #3
    thanks :)
     
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