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Thread: find an equation for surface of revolution

  1. #1
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    find an equation for surface of revolution

    I have a problem where i need to
    find the equation for surface of revolution if the generating curve y = 2x + 1, is revolved about the y axis


    The book gives this formula for revolving around the y axis

    $\displaystyle x^2 + z^2 = [ r(y)]^2$

    So, wouldnt i just plug y in and square it?
    most of the problems have been done this way

    Problem is i dont get the right answer.
    the answer is

    $\displaystyle 4x^2 -y^2 + 4z^2 + 2y - 1 =0$

    What do i need to do?
    Thank you
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  2. #2
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    Re: find an equation for surface of revolution

    The equation $\displaystyle x^{2}+z^{2}=[r(y)]^{2}$ is that of a pair of cones (or is it a single cone ?) with a common vertex and an axis of symetry along the $\displaystyle y$ axis. The cones open away from each other from the common vertex. The $\displaystyle r$ in the equation is the radius of the cone and the way that it is written $\displaystyle r(y)$ says that it is a function of $\displaystyle y.$ The radius will be zero when $\displaystyle y=1$ and $\displaystyle 1/2$ when $\displaystyle y=0,$ (or $\displaystyle y=2),$ these coming from the generating line $\displaystyle y=2x+1.$
    What you are looking for then is a relationship of the form $\displaystyle r=ay+b$ for some values of $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle r(0)=1/2$ and $\displaystyle r(1)=0.$
    That turns out to be $\displaystyle r=(1-y)/2$ and when you substitute that into the given equation it simplifies to your given answer.
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  3. #3
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    Re: find an equation for surface of revolution

    BobP, mathematically, its a single cone. Each part, that in "standard English" is called a cone, is a "nappe" of that cone.

    icelated, the "r(y)" is the function x= f(y) in the plane. Since you are given the line y= 2x+ 1, x= r(y)= (y- 1)/2. That is what should be squared, $\displaystyle x^2+ z^2= ((y- 1)/2)^2= (y^2- 2y+ 1)/4$ so that $\displaystyle 4x^2- y^2+ 4z^2- 2y= 1$.
    Thanks from icelated
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  4. #4
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    Re: find an equation for surface of revolution

    @HallsofIvy thank you that makes perfect sense.
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