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Math Help - arc length - are of the zone of a sphere by revolving about the y

  1. #1
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    arc length - are of the zone of a sphere by revolving about the y

    I have a problem where i dont understand how the solutions manual is doing the steps.

    Problem: Find the area of the zone of a sphere formed by revolving the graph of
    y = \sqrt 9 - x^2 ,  0< x < 2 about the y axis.

    so, y = \sqrt 9 - x^2

    then, sorry the square root suppose to go over the whole equation.

    y \prime =  \frac {-x} {\sqrt 9 - x^2}

    usually at this point it is 1 + (y \prime)^2

    The solutions manual shows

    \sqrt 1 + (y \prime)^2

    Why did they do this?

    Abd the next step is confusing..

     \frac {3}{\sqrt 9 - x^2}
    Where did the 3 come from?

    And in the next step where did the x come from?
    S = 2 \pi \int \frac {3x}{\sqrt 9 - x^2}
    Last edited by icelated; October 22nd 2011 at 04:22 PM.
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  2. #2
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    Clarksville, ARk
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    Re: arc length - are of the zone of a sphere by revolving about the y

    Quote Originally Posted by icelated View Post
    I have a problem where i don't understand how the solutions manual is doing the steps.

    Problem: Find the area of the zone of a sphere formed by revolving the graph of
    y = \sqrt 9 - x^2 ,  0< x < 2 about the y axis. I think you meant this to be y = \sqrt{9 - x^2}\, ,\   0<x<2

    so, y = \sqrt 9 - x^2

    then, sorry the square root suppose to go over the whole equation.

    y \prime =  \frac {-x} {\sqrt 9 - x^2}

    usually at this point it is 1 + (y \prime)^2

    The solutions manual shows

    \sqrt 1 + (y \prime)^2 Likewise, this should be \sqrt{ 1 + (y \prime)^2}

    Why did they do this?

    And the next step is confusing..

     \frac {3}{\sqrt 9 - x^2} Likewise, this should be  \frac {3}{\sqrt{ 9 - x^2}}

    Where did the 2 come from?

    And in the next step where did the x come from?
    S = 2 \pi \int \frac {3x}{\sqrt 9 - x^2} dx
    I corrected some of your LaTeX. Use { } with the \sqrt , as in \sqrt{9 - x^2}.

    Have you learned how to find arc length? ... or volume of a solid of revolution?

    Finding area for a surface of revolution, is based on finding arc length. The area for a surface of revolution is to arc length, as finding the volume of a solid of revolution is to using the integral to find area under a curve.
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