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Math Help - Integration with cylindrical shells

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    Integration with cylindrical shells

    Consider the given curves to do the following.
    Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about y = -4.

    I'm getting 302pi/60
    I set it up as 2pi*(the integral from 0 to 1) of (y-4)((y^2)-(y^.5))dy
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    Quote Originally Posted by Hellacious D View Post
    Consider the given curves to do the following.
    Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about y = -4.

    I'm getting 302pi/60
    I set it up as 2pi*(the integral from 0 to 1) of (y-4)((y^2)-(y^.5))dy
    V = 2\pi \int_0^1 (y+4)(\sqrt{y} - y^2) \, dy

    confirmed w/ washers ...

    V = \pi \int_0^1 (\sqrt{x}+4)^2 - (x^2+4)^2 \, dx<br />
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    Thank you, but I need to be able to do this with shells.
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    Quote Originally Posted by Hellacious D View Post
    Thank you, but I need to be able to do this with shells.
    I did it two ways ... the first integral is done with shells.
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    V = 2\pi \int_0^1 (y+4)(\sqrt{y} - y^2) \, dy
    Thanks, but can you explain how you knew to add 4 and subtract y^2? For some reason, I can't wrap my head around this method when I'm rotating about axes other than x or y.
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    Quote Originally Posted by Hellacious D View Post
    V = 2\pi \int_0^1 (y+4)(\sqrt{y} - y^2) \, dy
    Thanks, but can you explain how you knew to add 4 and subtract y^2? For some reason, I can't wrap my head around this method when I'm rotating about axes other than x or y.
    sketch a representative horizontal rectangle within the given region.

    the right end of the rectangle is x = \sqrt{y}

    the left end of the rectangle is x = y^2

    the height of the rectangle is (right - left) = \sqrt{y} - y^2 and the thickness of the rectangle is dy

    the rectangle is a distance of y above the x-axis

    the vertical distance from the rectangle to the line y = -4 is y - (-4) = y+4 , so the radius of revolution is y+4

    the volume of a single shell is ...

    dV = 2\pi(radius of revolution)(height of rectangle)(rectangle thickness)

    dV = 2\pi (y+4)(\sqrt{y} - y^2) \, dy
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