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Math Help - volume of a paraboloid

  1. #1
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    volume of a paraboloid

    I need help with determining the limits using cylindrical coordinates

    the volume is contained between the surface of a paraboloid z=4-x^2-y^2 and the plane z=0
    can anyone help me please??
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  2. #2
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    do i use the limits for theta 0 and 2pi

    for r 2 and 0 and for z 4 - r^2
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by calculusgeek View Post
    do i use the limits for theta 0 and 2pi

    for r 2 and 0 and for z 4 - r^2
    Yup.

    Thus, your integral would be \int_0^{2\pi}\int_0^2\int_0^{4-r^2}\,dz\,dr\,d\vartheta

    --Chris
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  4. #4
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    thanks very much

    i was wonder if you would know how to help me with changing to polar co ordinates in the double integral

    arctan (y/x) dxdy

    with the limit (x-1)^2+y^2<1
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  5. #5
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by calculusgeek View Post
    thanks very much

    i was wonder if you would know how to help me with changing to polar co ordinates in the double integral

    arctan (y/x) dxdy

    with the limit (x-1)^2+y^2<1
    In converting from rectangular to polar,

    (x-1)^2+y^2<1\implies 0<r<2\cos\vartheta and 0<\vartheta<\pi

    Thus, the double integral in polar coordinates will be \int_0^{\pi}\int_0^{2\cos\vartheta}\vartheta r\,dr\,d\vartheta, since \tan^{-1}\left(\frac{y}{x}\right)=\tan^{-1}\left(\frac{r\sin\vartheta}{r\cos\vartheta}\righ  t)=\tan^{-1}\left(\frac{\sin\vartheta}{\cos\vartheta}\right)  =\tan^{-1}\left(\tan\vartheta\right)=\vartheta

    --Chris

    p.s. From now on, ask new questions in a new thread
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