Results 1 to 4 of 4

Math Help - volume of rotation

  1. #1
    Junior Member
    Joined
    Mar 2007
    Posts
    63

    volume of rotation

    Using the shell method, find the volume of the solid of revolution obtained by rotating the region bounded by y = x and y = x2 about the x-axis
    <a> pi/2
    <b>2pi/5
    <c>pi/15
    <d>2pi/5
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by harry View Post
    Using the shell method, find the volume of the solid of revolution obtained by rotating the region bounded by y = x and y = x2 about the x-axis
    <a> pi/2
    <b>2pi/5
    <c>pi/15
    <d>2pi/5
    since we are rotating about the x-axis, we have to do it with respect to y.

    y = x
    => x = y

    y = x^2
    => x = sqrt(y)

    what are our limits of integration?

    they are where
    sqrt(y) = y
    => y = y^2
    => y^2 - y = 0
    y(y - 1) = 0
    => y = 0, y = 1 ......these are our limits.

    now see the graph below, we have to look at it sideways now.
    note that the height of the piece we want to rotate is the height of the higher graph minus the height of the lower graph. so height = sqrt(y) - y

    by the shell method
    V = int{(circumference)(height)}dy where circumference is 2pi*r, our radius here is just y

    so V = int{2pi*y*h}dy
    => V = int{2pi*y*(sqrt(y) - y)}dy
    => V = 2pi*int{y(y^(1/2) - y}dy
    => V = 2pi*int{y^(3/2) - y^2}dy
    => V = 2pi*[(2/5)y^(5/2) - (1/3)y^3] evaluated between 0 and 1, the limits we found above

    => V = 2pi*[2/5 - 1/3]
    => V = 2pi/15
    Attached Thumbnails Attached Thumbnails volume of rotation-shell.gif  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by harry View Post
    Using the shell method, find the volume of the solid of revolution obtained by rotating the region bounded by y = x and y = x2 about the x-axis
    <a> pi/2
    <b>2pi/5
    <c>pi/15
    <d>2pi/5
    for practice, try doing this by the disk (washer) method to verify the answer.

    washer method says:

    V = int{pi*(outer radius)^2 - pi*(inner radius)^2}dx


    P.S. you've called several threads "multiple choice" now. why not shake things up and call it something more exciting, like "AAAAHHHHH! HEELLPP!!! Shell Method! God Save Us All!"
    Follow Math Help Forum on Facebook and Google+

  4. #4
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    as always, if there's anything you don't get, say so, and myself or someone else will help you.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Volume of rotation (y-axis)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 1st 2010, 05:51 PM
  2. Calculating the Volume of a Rotation.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 12th 2010, 10:30 AM
  3. Volume by rotation
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 4th 2009, 07:00 AM
  4. Volume by rotation
    Posted in the Calculus Forum
    Replies: 4
    Last Post: September 15th 2008, 09:05 AM
  5. Area and rotation volume
    Posted in the Calculus Forum
    Replies: 8
    Last Post: June 12th 2008, 09:46 AM

Search Tags


/mathhelpforum @mathhelpforum