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Thread: Surface area problem

  1. #1
    Oct 2011

    Surface area problem

    I have a study guide with a couple problems like this, but I'll just post one and try to go from there. It isn't the actual integration that's giving me trouble but just the setting up of the integral.

    "Find the area of the following surface: the part of the paraboloid x=y^2+z^2 that lies inside the cylinder y^2+z^2=9"

    So i understand i have to parameterize somehow, and i understand the the surface area is the double integral of the magnitude of the cross product of the two partial derivatives of that parameterization. But I don't know how to set it up.
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  2. #2
    MHF Contributor

    Apr 2005

    Re: Surface area problem

    Yes, you have to parameterize the paraboloid surface. And looking at that $\displaystyle y^2+ z^2$, I would think of using "cylindrical coordinates" but with the polar coordinates part in the yz-plane rather than the xy-plane. That is, take x as a parameter and $\displaystyle \theta$ as the other parameter with $\displaystyle y= x^{1/2}cos(\theta)$, $\displaystyle z= x^{1/2}sin(\theta)$ so that $\displaystyle y^2+ z^2= (x^{1/2}cos(\theta))^2+ (x^{1/2}sin(\theta))^2= x$ as required.

    The condition that the surface be inside the cylinder $\displaystyle y^2+ z^2= 9$ just means that we must have $\displaystyle x= y^2+ z^2\le 9$. Your integral will be for $\displaystyle \theta$ from 0 to $\displaystyle 2\pi$, x from 0 to 9.
    Last edited by HallsofIvy; Nov 17th 2011 at 05:37 AM.
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