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Thread: Two ways of calculating area of a cylinder?

  1. #1
    May 2008

    Two ways of calculating area of a cylinder?

    Find the area of the part of the cylinder $\displaystyle x^2+y^2=1$ that is given by $\displaystyle 0<= z <= 2-x^2-2y^2$.
    (Hint: parametrize the surface.)

    I did manage to calculate the correct area by doing the following:
    Cylinder parametrization in xy-plane: r(t)=(x(t), y(t)) with x(t)=cos(t), y(t) = sin(t)

    Insert x(t) and y(t) in paraboloid equation to get area = $\displaystyle \int\limits_0^{2\pi } {(2 - \cos ^2 t - 2\sin ^2 t)dt}
    $ (ds of r(t) is 1*dt)

    How would you calculate the area by parametrizing the surface? For starters, how do you parametrize the surface?
    Last edited by kloda; May 13th 2008 at 05:38 AM.
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