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Math Help - Find the volume of the object using method of shells.

  1. #1
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    Find the volume of the object using method of shells.

    Hello all.

    My problem is this:

    Rotate the region bounded by the given curves about the line indicated. Obtain the volume of the solid by the method of shells. (give your answer in terms of .)

    x=sqrt(4-y^2) y-axis, x-axis. About the y-axis

    I've been working on this for hours and cannot get it. Help would be appreciated.
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  2. #2
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    When using shells, the cross sections are parallel to the axis we are revolving about. In this case, we are revolving about y so the cross sections will be parallel to the y-axis and 'stacked up' along the x-axis.

    So, integrate in terms of x.

    What we have here is a hemisphere of radius 2.

    Solving the given equation for y in terms of x gives:

    y=\pm\sqrt{4-x^{2}}

    Since we are bounded by the axes, we are in the positive region:

    2{\pi}\int_{0}^{2}x\sqrt{4-x^{2}}dx

    Check your result using the volume of a hemisphere formula. V=\frac{2}{3}r^{3}
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  3. #3
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    Quote Originally Posted by matt1989 View Post
    Hello all.

    My problem is this:

    Rotate the region bounded by the given curves about the line indicated. Obtain the volume of the solid by the method of shells. (give your answer in terms of .)

    x=sqrt(4-y^2) y-axis, x-axis. About the y-axis

    I've been working on this for hours and cannot get it. Help would be appreciated.
    note that the graph of this relation is a semicircle in quads I and IV

    the solid formed by rotating the graphed region about the y-axis will yield a sphere.

    what confuses me is the statement that the x-axis is a boundary for the region ... if so, then I suppose the region could either be the quarter circle in quad I or in quad IV.

    if that is the case, the volume of the solid hemisphere formed by using cylindrical shells is ...

    V = 2\pi \int_0^2 x \sqrt{4 - x^2} \, dx

    if the whole sphere is required, double the result.
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  4. #4
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    First of all thanks for the quick replys that was really helpful and I was able to complete the problem!

    Is it possible I could get help with 1 more problem? It is the same type of problem again but slightly different.

    It is the following:

    Use the method of shells to find the volume of the solid obtained by revolving the region bounded by the given curves about the - axis. (give your answer in terms of .)

    y= -x^2+13x-40 x=0 x=8 x-axis.

    I know that I have to break it into a left half and right half and then add the two integrals together but I'm not sure if I have them set up correctly.

    This problem seems extremly long the method that I've been going about it and I feel like I keep making mistakes some where. If I knew how to enter in all my work that I've done I would but I can't figure it out atm.

    Any help would be appreciated.
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  5. #5
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    Note that when we factor, we get x^{2}-13x+40=(x-5)(x-8)

    This tells us where to break up the integration limits.

    2{\pi}\int_{0}^{8}x|x^{3}-13x^{2}+40x|dx

    2{\pi}\left[\int_{0}^{5}x(x^{2}-13x+40)+\int_{5}^{8}x(-x^{2}+13x-40)\right]dx
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