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Math Help - volume by rotation about x axis

  1. #1
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    volume by rotation about x axis

    I need to obtain the volume of the region bound by y = x^2 y = 1 and the y axis by rotating it around the x axis and then about the y axis I could sure use some help if anyone has the time to explain how to attack this. Thank you
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  2. #2
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    Quote Originally Posted by Frostking View Post
    I need to obtain the volume of the region bound by y = x^2 y = 1 and the y axis by

    [1] rotating it around the x axis and

    [2] then about the y axis.

    I could sure use some help if anyone has the time to explain how to attack this. Thank you
    to [1]:

    The solid is a cylinder with radius r = 1 and height/length = 2 (from x = -1 to x = 1):

    V = \int_{-1}^1\left(\pi\cdot 1^2 - \pi x^2\right)dx


    to [2]:

    The solid is a kind of bowl with the parabola as cross-section:

    V=\int_0^1(\pi \cdot (\sqrt{y})^2)dy

    For your confirmation: The volumes are \dfrac12 \pi and \dfrac43 \pi .
    Attached Thumbnails Attached Thumbnails volume by rotation about x axis-rotparaboloid_xaxis.png   volume by rotation about x axis-rotparaboloid_yaxis.png  
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