Results 1 to 12 of 12
Like Tree1Thanks
  • 1 Post By MarkFL

Math Help - Find the volume of the solid obtained by rotating the region bounded by the curves...

  1. #1
    Newbie
    Joined
    Sep 2012
    From
    Philadelphia
    Posts
    10

    Find the volume of the solid obtained by rotating the region bounded by the curves...

    The region bounded by: y=1 and y=x^8
    Rotated around: y=4

    I used to know how to do these, but I can't remember it appears and after about an hour of trying to figure it out by trying new things I've given up. I really just need the correct integral that comes out of it I suppose, not an actual number answer.

    Thank you for your help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Find the volume of the solid obtained by rotating the region bounded by the curve

    I would use the washer method here, although the shell method is also fairly straightforward here. A good place to begin is with the formula for the volume of a washer, in terms of the outer radius, the inner radius and the thickness. What is this formula?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2012
    From
    Philadelphia
    Posts
    10

    Re: Find the volume of the solid obtained by rotating the region bounded by the curve

    Integral from a to b of pi*(((outer)^2)-((inner)^2))

    From what I can tell, the inner radius would simply be 6, no? And the outer would be 1-(x^8)?

    I would believe it's from -1 to 1, but that doesn't seem to be working. I've tweaked here and there...but I can't find my mistake.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Sep 2012
    From
    Philadelphia
    Posts
    10

    Re: Find the volume of the solid obtained by rotating the region bounded by the curve

    Inner is 3...silly me...hang on.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Sep 2012
    From
    Philadelphia
    Posts
    10

    Re: Find the volume of the solid obtained by rotating the region bounded by the curve

    No beyond that I'm still missing something haha.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Find the volume of the solid obtained by rotating the region bounded by the curve

    You are on the right track...the volume of a washer is:

    V_w=\pi\left(r_o^2-r_i^2 \right)h

    where r_o is the outer radius, r_i is the inner radius and h is the thickness.

    If we slice our solid of revolution vertically, i.e., perpendicular to the x-axis, the thickness of each slice we may call dx.

    You are right that the inner radius is 3 units, since this is the distance between the upper bound of the region being rotated and the axis of rotation.

    So what is the distance between the lower bound of the region being rotated and the axis of rotation?
    Last edited by MarkFL; September 16th 2012 at 07:54 PM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Sep 2012
    From
    Philadelphia
    Posts
    10

    Re: Find the volume of the solid obtained by rotating the region bounded by the curve

    Isn't it just the distance between y=1 and y= x^8? And seeing as y=1 is on top, it would be that minus y=x^8?
    Last edited by ScareKRO; September 16th 2012 at 07:55 PM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Find the volume of the solid obtained by rotating the region bounded by the curve

    No, you want the distance between the axis of rotation y=4 and the lower bound of the rotated region, which is y=x^8.

    What is this distance?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Find the volume of the solid obtained by rotating the region bounded by the curve

    By the way, an invaluable aid to doing these problems is to sketch the graphs of the bounding functions and the axis of rotation. Have you made such a sketch?
    Thanks from ScareKRO
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Newbie
    Joined
    Sep 2012
    From
    Philadelphia
    Posts
    10

    Re: Find the volume of the solid obtained by rotating the region bounded by the curve

    So then it would be (4-(x^8)). And yeah...I've drawn it about 100 times now haha.

    So then.....Integral of pi[((4-(x^8))^2)-(9)]
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Newbie
    Joined
    Sep 2012
    From
    Philadelphia
    Posts
    10

    Re: Find the volume of the solid obtained by rotating the region bounded by the curve

    Okay, I just got it to work.

    I see my mistake. I wasn't taking the outer radius, just the distance between the two given bounds...which is not how you do this haha. Now I just feel silly.

    Thank you very much for your help.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Find the volume of the solid obtained by rotating the region bounded by the curve

    I like to state the volume of an arbitrary slice as:

    dV=\pi\left(\left(4-x^8 \right)^2-3^2 \right)\,dx

    Since the bounded region is symmetric across the y-axis, we may then state (after determining the limits as -1 and 1):

    V=2\pi\int_0^1\left(\left(4-x^8 \right)^2-3^2 \right)\,dx

    Then, of course expand the binomial within the integrand and integrate term by term.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: March 9th 2011, 06:15 AM
  2. Replies: 1
    Last Post: February 11th 2010, 06:01 PM
  3. Replies: 7
    Last Post: October 23rd 2009, 03:43 AM
  4. Replies: 3
    Last Post: August 14th 2009, 07:34 AM
  5. Replies: 2
    Last Post: December 4th 2008, 04:30 PM

Search Tags


/mathhelpforum @mathhelpforum