1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Finding/sketching volume with the washer method

  1. Mar 17, 2008 #1
    [SOLVED] Finding/sketching volume with the washer method

    1. The problem statement, all variables and given/known data
    Consider the region bounded by the curves [tex] y=x^2+1[/tex] & [tex]y=3-x^2[/tex]. a) Using the disk/washer method, find the volume of the solid obtained by rotating this region about the x-axis. b) Setup the integral for finding the volume of the solid obtained by rotating this region about the x-axis. In each case, draw a diagram, indicating a typical "strip" (and its slice).

    2. Relevant equations

    3. The attempt at a solution
    I'm pretty sure the first one is right wrt x.

    Is this the correct way to set it up wrt y? Is there a simpler way?

    Also, what should I draw to "indicate a 'strip'"? I see the words strip, slab, and slice often used in a seemingly interchangeable fashion. What are they asking for here? Thanks for reading.
  2. jcsd
  3. Mar 17, 2008 #2
    Hello there.

    Your setup wrt has one mistake; It should be: [tex]\pi\int{[(3-x^2)^2-(x^2+1)^2]dx}[/tex]

    As for the second part, ideally you would not want to set this up wrt y, for the following two reasons:

    1) It changes your method for finding volume from the washer method to cylindrical shells (assuming that your "slice" is perpendicular to the y-axis).

    2) Given my assumption about your slice in #1, you'd have to set up 3 integrals in order to do the problem, which is pretty insane.

    As for your last question, a "slice" or "strip" is simply the distance from one curve to the other for any given value of your variable of integration. For instance, take the point x=0.5. Draw a line or a narrow rectangle from the curve [tex]y=3-x^2[/tex] to the curve [tex]y=x^2 + 1[/tex] at that point. That is your slice/strip. The width of that strip is [tex]dx[/tex].
    Last edited: Mar 17, 2008
  4. Mar 17, 2008 #3
    Thanks for the reply. I believe I have corrected my answer wrt x based on your suggestion although the new answer of 16pi seems a little bit large for the size of the rotated region. Also, I'm still not sure how to set it up wrt y. I know it's the harder way to do it but I would prefer to learn (and the question does ask for it). Cheers.
    Last edited: Mar 17, 2008
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook