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Math Help - Volume of a cone

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    MHF Contributor arbolis's Avatar
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    Volume of a cone

    I'm having an extremely hard time trying to calculate the volume of the cone given by (z-1)^2=\frac{x^2}{2}+y^2, 0\leq z \leq 1.
    I've tried to draw it in the xyz plane : its generators cross when z=1, x=y=0. In the xy plane its cross section is an ellipse centered at the origin, whose semi major axis is worth 1 unit and its semi minor axis is worth 1/2. But I'm not sure I've done it right. Also I have a big problem : when using cylindrical coordinates, \theta goes from 0 to 2\pi, z goes from 0 to 1 but r goes from 0 to ... I'm unable to find it. I guess I must find some equation of generators, but still, the cone is not circular (cross section with the xy plane), making it difficult for me to find its volume.
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    Quote Originally Posted by arbolis View Post
    I'm having an extremely hard time trying to calculate the volume of the cone given by (z-1)^2=\frac{x^2}{2}+y^2, 0\leq z \leq 1.
    I've tried to draw it in the xyz plane : its generators cross when z=1, x=y=0. In the xy plane its cross section is an ellipse centered at the origin, whose semi major axis is worth 1 unit and its semi minor axis is worth 1/2. But I'm not sure I've done it right. Also I have a big problem : when using cylindrical coordinates, \theta goes from 0 to 2\pi, z goes from 0 to 1 but r goes from 0 to ... I'm unable to find it. I guess I must find some equation of generators, but still, the cone is not circular (cross section with the xy plane), making it difficult for me to find its volume.
    since z \leq 1, we get z=1-\sqrt{\frac{x^2}{2}+y^2}. the volume is V=\int \int_D \left(1-\sqrt{\frac{x^2}{2}+y^2} \ \right) \ dA, where D=\{(x,y) \in \mathbb{R}^2: \ \frac{x^2}{2}+y^2 \leq 1 \}. put x=\sqrt{2}r \cos \theta, \ y=r \sin \theta, \ 0 \leq \theta \leq 2 \pi, \ 0 \leq r \leq 1.

    under this transformation, D is transformed to the unit disc and the Jacobian is \sqrt{2}r. thus: V=\sqrt{2} \int_0^{2 \pi} \int_0^1 r(1-r) \ dr d \theta = \frac{\pi \sqrt{2}}{3}.
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