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Homework Help: Cylindrical shells question

  1. Jul 18, 2010 #1
    1. From Stewart Calculus and Concepts 4th edition, page 454 #15

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

    15. y=4x-x^2, y=; rotate about x=1

    2. Relevant equations



    3. I was able to find the volume of the function y=4x-x^2 by integrating from 0 to 4, but the line y=3 cuts the function off near the top, and I don't know how to take some of that volume "out". Are 2 integrals needed for this? I also tried the washer method of integrating the top function(y=4x-x^2 minus the bottom function(y=3) and I still didn't get it. Ive been stuck on this one for about 2 hours now. Any hints?
     
  2. jcsd
  3. Jul 18, 2010 #2
    Nope, we don't need 2 integrals. Since this is rotation about a vertical axis (x=1), we have

    [tex] V = \int_a^b 2\pi*r*h*dx [/tex]
    Where,
    r = distance from axis
    h = height of shell.

    That's all we need. As for hints:
    Since we're rotating about x=1, we're essentially "losing" 1 unit of x in the sense that r = x - 1.
    Also, h is the difference of the two y(x) functions (top minus bottom).

    Can you figure out the rest? I think you should end up with V = 4/3.
     
  4. Jul 18, 2010 #3
    The back of the book says the answer is 8*pi/3

    I know how to do some of these but this one I must just be missing something fundamental.

    I don't know how to do a definite integral, but its from 0 to 4 of[tex]2\pi\int(x-1)(4x-x^2-3)dx[/tex]
     
  5. Jul 18, 2010 #4
    ooops....yea, I forgot the factor of 2pi out in front. So 4/3 * 2pi = 8pi/3 is correct.

    Close!!! Everything inside the integral is good, but your limits are a bit off, i.e. NOT from 0 to 4. Remember, the limits of your integral will be the 2 x-values that bound the region. Between which two x-values does the region lie?
     
  6. Jul 18, 2010 #5
    all I know is the functions intersect at x=1 and x=3. Are those the limits? If so then I don't understand why.
     
  7. Jul 18, 2010 #6
    Okay I got the answer finally.

    The whole time I was trying to solve it I was thinking for some reason that they were looking for the bottom half of the curve, not the top part bisected by y=3. O well, lesson learned not to make blind assumptions! Thank you Raskolnikov
     
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