Results 1 to 2 of 2

Math Help - Parametric Curves and Volume of the area rotated about y-axis

  1. #1
    Junior Member
    Joined
    May 2011
    Posts
    30

    Parametric Curves and Volume of the area rotated about y-axis

    Hi guys! Need help with this question please,

    The curve C is defined parametrically by x = (1+t)^(2/3), y = lnt^2, t < than or = to -1
    Find the exact volume generated when the area of the region enclosed by C, the lines x = 0, x = 1 and the x-axis is rotated through 2pi radians about the y-axis.

    This is my working:
    Req. vol. = Vol. of cylinder - Vol. of curve rotated about y axis from y = 0 to y = ln4
    (y = ln4 is the intersection b/w C and the line x = 1)
    Vol. of cylinder = pi(r^2)(h) = pi(1^2)(ln4) = (pi)(ln4)
    For vol. of the curve,
    y = ln(t^2) => t = e^(y/2)
    Subst. into x = (1+t)^(2/3), x = [1+e^(y/2)]^(2/3)
    Vol. of curve = pi * (integrate x^2 dy from y = 0 to y = ln4)
    where x^2 = [1+e^(y/2)]^(4/3)

    However, the vol. of curve found is much bigger than the vol. of cylinder. What went wrong?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    12,041
    Thanks
    885

    Re: Parametric Curves and Volume of the area rotated about y-axis

    Quote Originally Posted by Blizzardy View Post
    Hi guys! Need help with this question please,

    The curve C is defined parametrically by x = (1+t)^(2/3), y = lnt^2, t < than or = to -1
    Find the exact volume generated when the area of the region enclosed by C, the lines x = 0, x = 1 and the x-axis is rotated through 2pi radians about the y-axis.

    This is my working:
    Req. vol. = Vol. of cylinder - Vol. of curve rotated about y axis from y = 0 to y = ln4
    (y = ln4 is the intersection b/w C and the line x = 1)
    Vol. of cylinder = pi(r^2)(h) = pi(1^2)(ln4) = (pi)(ln4)
    For vol. of the curve,
    y = ln(t^2) => t = e^(y/2)
    Subst. into x = (1+t)^(2/3), x = [1+e^(y/2)]^(2/3)
    Vol. of curve = pi * (integrate x^2 dy from y = 0 to y = ln4)
    where x^2 = [1+e^(y/2)]^(4/3)

    However, the vol. of curve found is much bigger than the vol. of cylinder. What went wrong?
    you are using cross-sections of disks ... did you look at the graph of the curve? Seems the method of washers is needed to rotate the described region about the y-axis.

    also, note that since t \le -1 ...

    t^2 = e^y

    t = -e^{y/2}

    so, x = (1 - e^{y/2})^{2/3}


    V = \pi \int_0^{\ln{4}} 1 - (1 - e^{y/2})^{4/3} \, dy
    Attached Thumbnails Attached Thumbnails Parametric Curves and Volume of the area rotated about y-axis-parametric1.jpg  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] find the volume of the curves rotated about the x-axis
    Posted in the Calculus Forum
    Replies: 3
    Last Post: July 11th 2010, 12:46 PM
  2. Volume of a solid rotated aroun the x axis
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 19th 2010, 05:53 PM
  3. volume of a solid rotated about the y-axis
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 1st 2010, 02:18 PM
  4. Volume for region rotated around the x-axis
    Posted in the Calculus Forum
    Replies: 5
    Last Post: March 30th 2009, 12:11 AM
  5. Volume of a curve rotated around the y-axis
    Posted in the Calculus Forum
    Replies: 3
    Last Post: July 29th 2008, 12:36 AM

Search Tags


/mathhelpforum @mathhelpforum