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Math Help - Surface of the cone inside the cylinder

  1. #1
    Newbie Metrika's Avatar
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    Question Surface of the cone inside the cylinder

    Help me compose the surface integral to solve this problem, please

    Calculate the area of the surface of the cone x^2-y^2-z^2=0, located inside the cylinder x^2+y^2=1.
    Make a drawing.
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  2. #2
    Senior Member DeMath's Avatar
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    Quote Originally Posted by Metrika View Post
    Help me compose the surface integral to solve this problem, please

    Calculate the area of the surface of the cone x^2-y^2-z^2=0, located inside the cylinder x^2+y^2=1.
    Make a drawing.
    See drawing; but I do not know how to compose the integral to your problem((

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  3. #3
    Newbie Metrika's Avatar
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    omg! stunningly beautiful!
    In what program this is done?

    Is WolframAlpha?
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  4. #4
    MHF Contributor

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    So that we can write this in regular polar cylindrical coordinates, I am going to swap "x" and "z" in the problem:
    z^2- x^2- y^2= 0 and z^2+ y^2= 1.

    In cylindrical coordinates, that is z^2= r^2 and z^2+ r^2 sin^2(\theta)= 1 so the two surfaces intersect where r^2+ r^2 sin^2(\theta)= 1 or r= \pm \sqrt{\frac{1}{1+ sin^2(\theta)}. Integrate with \theta from 0 to 2\pi and, for each \theta, r from 0 to \frac{1}{\sqrt{1+ sin^2(\theta)}}.

    The "differential of surface area" for the cone is \sqrt{2} r drd\theta.
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  5. #5
    Senior Member DeMath's Avatar
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    Quote Originally Posted by Metrika View Post
    omg! stunningly beautiful!
    In what program this is done?

    Is WolframAlpha?
    I did it with the Maple_13; нere's the code for building


    A := plot3d([[r*cos(t),r*sin(t),r*sqrt(cos(2*t))],[r*cos(t),r*sin(t),-r*sqrt(cos(2*t))]], t=-(1/4)*Pi .. (1/4)*Pi, r=0 .. 1, numpoints=3000, color="LightBlue", style=surface):
    B := plottools[rotate](A,0,0,Pi):
    C := plot3d([cos(t),sin(t),z], t=0 .. 2*Pi, z=-1 .. 1, color=pink, style=wireframe, transparency=0.75, numpoints=2000):
    plots[display](A,B,C, axes=normal, scaling=constrained, lightmodel=light2, view=[-1.4 .. 1.4, -1.4 .. 1.4, -1.4 .. 1.4], orientation=[70,62]);
    Last edited by DeMath; November 27th 2010 at 10:05 AM.
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