Results 1 to 12 of 12

Math Help - shell method

  1. #1
    Member
    Joined
    Jan 2011
    Posts
    88

    shell method

    I'm having trouble with finding the volume of y=x^2 from x=0, to x=1, revolving about the x axis. I'm having trouble visualizing it with this method. I can execute it with other methods, but not shell.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by quantoembryo View Post
    I'm having trouble with finding the volume of y=x^2 from x=0, to x=1, revolving about the x axis. I'm having trouble visualizing it with this method. I can execute it with other methods, but not shell.
    Using the washer method, it's easy to visualise them by comparing them to the rings of Saturn.

    For the shells, pick a point on the y-axis at about 0.8
    and draw a horizontal line until it touches the curve above the positive x-axis.
    Rotate this line about the x-axis.
    It traces out a cylinder.
    You want the curved surface area of this cylinder (the surface area of the shell).
    Now imagine doing the same for all the other horizontal lines from y=0 to y=1.
    Integrate all the surface areas to get the volume (like layers of an onion).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jan 2011
    Posts
    88
    When I envision what you say, it comes up wrong. The shell will have dimensions 2pi*f(x), x, and dxm I get dV=2pi*x^3dx which can't be right. Help?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by quantoembryo View Post
    When I envision what you say, it comes up wrong. The shell will have dimensions 2pi*f(x), x, and dx.

    I get dV=2pi*x^3dx which can't be right. Help?
    Looking more closely at the cylinders.
    The "thickness" of the cylinders.... is it "dy" or "dx".
    Over which axis do we integrate?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Jan 2011
    Posts
    88
    I don't know what my issues are with this.... I can now see that it is dy, but that's about it. I keep ending up with 2pi*x^3 and I know that's wrong. My textbook gives one example and I just don't get it..
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by quantoembryo View Post
    I don't know what my issues are with this.... I can now see that it is dy, but that's about it. I keep ending up with 2pi*x^3 and I know that's wrong. My textbook gives one example and I just don't get it..
    Yes, you are integrating over the vertical axis.
    Hence you need f(y) instead of f(x).

    y=f(x)=x^2\Rightarrow\ x=\sqrt{y}

    This gives the cylinder heights.

    2{\pi}rh=2{\pi}yx=2{\pi}y\sqrt{y}
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Jan 2011
    Posts
    88
    Alright, I integrated that and ended up with 4pi/5, however, I did it via splices and got pi/5 which I know is right
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by quantoembryo View Post
    When I envision what you say, it comes up wrong. The shell will have dimensions 2pi*f(x), x, and dxm I get dV=2pi*x^3dx which can't be right. Help?
    Yes, you've calculated the VOR for the region above the curve.

    You need the region below the curve.
    Hence, the height of each cylinder is 1-x.
    Attached Thumbnails Attached Thumbnails shell method-vor-x-2.jpg  
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Jan 2011
    Posts
    88
    Thanks a lot for the help. However, my final concern is I don't see why the height is 1-x rather than just x, even from your diagram
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by quantoembryo View Post
    Thanks a lot for the help. However, my final concern is I don't see why the height is 1-x rather than just x, even from your diagram
    All of the cylinders have to rest against the line x=1.
    This is the case when we are calculating the VOR of the region
    between the curve and horizontal axis using shells.
    I've included 2 such cylinders, though there are "infinitely" many.
    Therefore we need to subtract x from 1 to get the cylinder heights,
    since the lines of length 1-x are being rotated around the horizontal axis.

    Hope this helps.

    Remember, these cylinders are "resting on their sides", not standing vertically.
    Attached Thumbnails Attached Thumbnails shell method-vor.2-x-2.jpg  
    Last edited by Archie Meade; February 25th 2011 at 04:36 AM.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member
    Joined
    Jan 2011
    Posts
    88
    Yes, I finally am understanding. Thanks for your patience!
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Thanks, sorry for not distinguishing between the region above the curve under the imaginary line y=1
    and the region below the curve above the horizontal axis earlier in the thread.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Shell Method Help
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 14th 2010, 12:50 PM
  2. Replies: 5
    Last Post: January 22nd 2010, 06:50 AM
  3. Finding the shell height in the Shell Method
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 26th 2009, 02:47 PM
  4. Shell Method
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 29th 2008, 12:57 PM
  5. Replies: 2
    Last Post: August 17th 2008, 01:02 PM

Search Tags


/mathhelpforum @mathhelpforum