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Math Help - Change of variable triple integral question.

  1. #1
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    Change of variable triple integral question.

    Question asks:

    Find the volume of the solid bounded by: z = 4x^2 + y^2 and the cylinder y^2+z=2

    In the solution they used change of variable by putting in x = r cosθ/sqrt 2 and
    y = r sin θ, z = z

    this looks like cylindrical co ordinates to me, but I thought that you always use
    x = r cos θ

    can someone explain why divide it by root 2? And how can you just look at questions like these and decide what change of variable to make? I'm really confused.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Change of variable triple integral question.

    Quote Originally Posted by Kuma View Post
    this looks like cylindrical co ordinates to me, but I thought that you always use x = r cos θ
    The projection onto the xy plane of the intersection of the surfaces z=4x^2+y^2 and y^2+z=2 is 4x^2+2y^2=2 or equivalently \frac{x^2}{(1/\sqrt{2})^2}+y^2=1 (ellipse). The substitution x=(r\cos \theta)/\sqrt{2},\;y=r\sin \theta transforms the ellipse into the circle r=1 .
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