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Math Help - Rotated parametric curve : calculate the volume

  1. #1
    MHF Contributor arbolis's Avatar
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    Rotated parametric curve : calculate the volume

    I'm unsure of the way to approach the problem : Let C:[0,\pi]\to \mathbb{R}^3 be the curve given by C(t)=(0,\sin t , t). Calculate the volume enclosed by the curve if we rotate C with respect to the z axis.
    I think a way would be to determine the solid and then evaluate the volume via a triple/double or even simple integral.
    I wonder if there's an easier way to solve the problem.
    Otherwise I'm not sure what solid the rotated curve represent. An ellipsoid?
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  2. #2
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    Quote Originally Posted by arbolis View Post
    I'm unsure of the way to approach the problem : Let C:[0,\pi]\to \mathbb{R}^3 be the curve given by C(t)=(0,\sin t , t). Calculate the volume enclosed by the curve if we rotate C with respect to the z axis.
    I think a way would be to determine the solid and then evaluate the volume via a triple/double or even simple integral.
    I wonder if there's an easier way to solve the problem.
    Otherwise I'm not sure what solid the rotated curve represent. An ellipsoid?
    Hey, ain't that \pi r^2 dx long-ways? I mean up the z-axis. The parametric representation you gave is just a sine wave along the y-z axis in the form of y=\sin(z). So takin' disks, that's just \pi \sin^2(z) dz from 0 to \pi. Think so anyway.
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  3. #3
    MHF Contributor arbolis's Avatar
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    Thanks! Ok so if I understood well, V=\pi \int_0^{\pi} \sin ^2 (z)dz = \frac{\pi ^2}{2}.
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