Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis.

  1. 1. The region bounded by the given curves is rotated about the specific axis. Find the volume of the resulting solid by any method (disc or shell).



    2. x^2 + (y-1)^2 = 1



    3. This is my first time posting up a homework question, so I apologize if I didn't get the notation down correctly or something.

    I tried shells and ended up with ∏ as my answer. (I hope that's a pi symbol.)

    V= 2(2∏radius)(width) I doubled the circumference to get the whole circle rather than just having the semi-circle.

    V = 4∏ ∫y sqrt[ 1- (y-1)^2] dy on [0,2]
     
    Last edited by a moderator: Mar 15, 2012
  2. jcsd
  3. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    The two dimensional figure is definitely a circle which passes through the origin. The rotation about x axis would make it a horn torus (that's a funny name actually).

    Using the formula that I remember (it's derived using integration, plus looks obvious if you see)
    V = (π(r^2))(2πR)

    Now putting r=1 and R=(1/2)

    V=(π(1))(π)

    So my answer would be π^2

    I might me miserably wrong though.
     
  4. Mark44

    Staff: Mentor

    Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    You doubled the width from x = 0 to x = +√(1 - (y - 1)2).

    Also, your volume formula needs something for the thickness of the shell, which in this case is Δy.
     
    Last edited: Mar 15, 2012
  5. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    I got [0,2] in regards to the y-axis, not the x-axis, which is [0,1]. My mistake, sorry!

    I'll retry what I did but on [0,1] this time.
     
  6. Mark44

    Staff: Mentor

    Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    No, [0, 2] is the right interval along the y-axis. I took that comment out right after I posted it, but I was a bit too slow.
     
  7. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    Oh, okay. I'm not sure what action I should take now. But was my mistake the doubling part?

    Actually, when I integrate there's one part in which I do substitution and the upper limit and lower limit became the same, so that part of my integral equaled 0. I thought that maybe that accounted for the "empty space". Maybe I should type out my whole solution...I'll do that now.
     
  8. Mark44

    Staff: Mentor

    Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    Yeah, show what you did. The integral is not amenable to an ordinary substitution, I don't believe.

    BTW, wolfram alpha says the value is 2π2, so I'm not sure I trust cng99's formula.
     
  9. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    V = 4∏ ∫ y sqrt[1-(y-1)^2] dy on [0,2]

    Substitution: u = y - 1 → du = dy

    V = 4∏ ∫ (u+1) sqrt(1-u^2) du on [-1,1]

    V = 4∏ ∫u sqrt(1-u^2) du + 4∏ ∫ sqrt(1-u^2) du on [-1,1]

    The left integral:

    Substitution: t = 1-u^2 → dt = -2du

    V = -2∏ ∫ sqrt(t) dt on [0,0] so that just equals 0

    The right integral:

    Substitution: u = sin∅ → du = cos∅ d∅

    V = 4∏ ∫ sqrt[1-sin^2∅] cos∅ d∅ on [arcsin(-1),arcsin(1)]

    V = 4∏ ∫ cos^2∅ d∅ on [arcsin(-1),arcsin(1)]

    V = 4∏ ∫ 1/2 [1+cos2∅] d∅ on [arcsin(-1),arcsin(1)]

    V = 4∏ [ 1/2∅ + 1/4 sin2∅] on [arcsin(-1),arcsin(1)]

    V = 4∏ [ 1/2 * ∏/2 + 1/4 * 0] - 4∏ [1/2 * 3∏/2 + 1/4 * 0 ]

    V = 4∏ [ ∏/4 - 3∏/4] (Whoops, I just realized my arithmetic mistakes!)

    V = -2∏

    Okay...so sorry about that. ∏ was not my original answer! And this is definitely wrong since it's negative....
     
    Last edited: Mar 15, 2012
  10. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    Yeah, I don't understand it at all.

    And ohhh, so π is pi!
     
  11. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    So...does anyone have advice for my solution?
     
  12. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    What's the answer given in the book?
     
  13. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    There's no answer provided in the book since this is an even numbered problem. But someone on a different forum told me the answer should be 2π^2. He used Theorem of Pappus, though, and I'm limited to discs and shells.
     
  14. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    Of course!!!
    It's 2π^2

    Volume = (2πR)(πr^2)

    http://whistleralley.com/torus/torus.htm

    And in this case R=1, r=1.

    Why do you wanna do integration?
     
  15. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    The problem's instructions says that I can only use discs or shells, so I have to use integration. Pappus is off limits, especially since I never learned it.
     
  16. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    Bump......
     
  17. Mark44

    Staff: Mentor

    Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    I don't have time to go through this right now, but I don't see anything obviously wrong in your approach. One suggestion is to just work with indefinite integrals (no limits of integration), and then evaluate your final antiderivative at the original limits of integration, 0 and 2. It could be that a small error in switching several times to different limits of integration might have caused your error.

    BTW, bumping a thread before 24 hours have passed is not permitted here at PF.
     
  18. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    Why don't you just use the disk method (well washer I guess) on the top portion of the arc.
    For example:

    X2+(y-1)2=1 Becomes:

    y=+/-sqrt(1-x2) + 1

    We just revolve the top arc around y=1 and multiply by 2.

    So:

    2[itex]\pi[/itex]∫[itex]^{1}_{-1}[/itex] R(x)2 - r(x)2dx

    R(x) = sqrt(1-x2) + 1
    r(x) = 1 (revolving about y=1)

    R(x)2-r(x)2 = (1-x2) + 2 sqrt(1-x2)
    So now we have:

    2[itex]\pi[/itex]∫[itex]^{1}_{-1}[/itex] (1-x2) + 2sqrt(1-x2)dx

    Splitting that into two integrals you get 8[itex]\pi[/itex]/3 for the first and
    the next integral 2[itex]\pi[/itex]∫[itex]^{1}_{-1}[/itex] 2sqrt(1-x2)dx needs to be done with trig substitution...
     
    Last edited: Mar 15, 2012
  19. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    For the second integral, I get 2π^2 for my answer...but then I have 8π/3 for the first integral...

    Oh, and sorry about the bump.
     
    Last edited: Mar 15, 2012
  20. Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    I took your advice and solved the indefinite integral and then I computed the original limits of integration. It came out to 2pi^2! Thanks a lot!
     
  21. Mark44

    Staff: Mentor

    Re: Find the volume of the region bounded by x^2+(y-1)^2 = 1 rotated about the x-axis

    There's nothing wrong with doing all the integrals as definite integrals, and transforming the limits of integration according to the substitutions being used. It just seems more prone to getting fouled up by arithmetic errors is all.
     
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