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Homework Help: Shell Method (Volume)

  1. Sep 3, 2006 #1
    Please help! How do I do this problem?

    Using the method of cylindrical shells, find the volume generated by rotating the region the region bounded by the given curves about the specified axis.

    y=(x-1)^(1/2), y=0, x=5; about y = 3

    Please tell me how to set up the integral! Any help is MUCH appreciated.

    So far I have Integral from 0 to 2 of (3-(y^2+1))*y dy I know that isn't right, because I am not getting the right answer! The book says that it is 24pi.

    Thank you.
  2. jcsd
  3. Sep 3, 2006 #2


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    Science Advisor

    Do you know what each shell looks like?

    The cross section of each shell is a long, narrow, flat rectangle, whose long dimension is parallel to the x-axis and stretches from the point where x = y^2 + 1 to the point where x = 5. It is parametrized by y. The shell itself is that rectangle rotated around the line y = 3. Ignore the small vertical width of the rectangle for a minute (which is dy) so that the shell is just like a horizontal tube with no thickness. What is the area of the outer surface of this tube? It has a radius--the radius is 3 - (y^2 + 1). So what is its circumference? What is its length?
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