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Math Help - Volumes question using the shells method

  1. #1
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    Volumes question using the shells method

    Heres the diagram (S is the yellow region)

    http://i249.photobucket.com/albums/g...AAAAAAATHH.gif

    Ok so the question is

    Lets S be the solid obtained by rotating the region shown in the figure about the y-axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of S. Do you think this method is preferable to slicing? Explain

    This question was set as home work today but the problem is we only started this topic today and I don't understand the topic to the point where i can make sense of all this. However i think i know how to sketch it. Sort of like a camels hump with a cylinder going down the center. So if you help me thanks a lot in advance! =)
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  2. #2
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    height of shell ... \sin(x^2)

    circumference of shell ... 2\pi x

    shell thickness ... dx

    V = \int_0^{\sqrt{\pi}} 2\pi x \cdot \sin(x^2) \, dx

    V = \pi \int_0^{\sqrt{\pi}} 2x \cdot \sin(x^2) \, dx

    now you may integrate using the method of substitution ...

    V = \pi \int_0^{\pi} \sin(u) \, du = 2\pi
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