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Math Help - Volume of given solid

  1. #1
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    Volume of given solid

    Find the volume of the given solid.
    Bounded by the cylinder y^2 + z^2 = 4 and the planes x = 2y, x = 0, z = 0 in the first octant
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  2. #2
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    Quote Originally Posted by purplerain View Post
    Find the volume of the given solid.
    Bounded by the cylinder y^2 + z^2 = 4 and the planes x = 2y, x = 0, z = 0 in the first octant
    Because of the obvious cylindrical symmetry, I would recommend changing to polar coordinates in the yz-plane. Then y= r cos(\theta), z= r sin(\theta) and the cylinder, y^2+ z^2= 4 becomes r= 2 and x= 2y gives height x= 2r cos(\theta).
    Integrate 2r cos(\theta) r dr d\theta with r ranging from 0 to 2 and \theta ranging from 0 to \pi/2.

    (Normally, of course, we use polar coordinates in an xy-plane. Essentially what I did was swap x and z so the problem would be "find the area of the solid bounded by x^2+ y^2= 4 in the first quadrant from z= 0 to z= 2y. Changing the coordinate system doesn't change the solid so the two solids have the same volume)
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