Results 1 to 4 of 4

Math Help - Volume by slices and shells

  1. #1
    Super Member angel.white's Avatar
    Joined
    Oct 2007
    Posts
    723
    Awards
    1

    Volume by slices and shells

    Question: Let V be the volume of the solid obtained by rotating about the y-axis the region bounded by y=\sqrt{x} and y=x^2. Find V both by slicing and by cylindrical shells. In both cases, draw a diagram to explain your method.


    My work:
    -------------------
    a) Volume by slices.
    Since this is rotated about the y-axis, I solved in terms of x:
    y=\sqrt{x} ~~~~\Rightarrow ~~~~x=y^2
    y=x^2 ~~~~~\Rightarrow ~~~~x = \sqrt{y}

    Then I said the cross-sectional areal will be \pi x^2 since x will be the radius.

    So I plugged my equations into it:
    Area_1 = \pi y
    Area_2 = \pi y^2

    Since they intersect at the points (0,0) and (1,1) Volume will be from y=0 to y=1:
    V=\int_0^1 (Area_1-Area_2)~dy

    V=\pi \int_0^1 (y-y^2)~dy

    V=\pi (\frac 12y^2-\frac 13y^3)_0^1

    V=\pi (\frac 12-\frac 13)

    V=\frac 16\pi

    -----------
    b) volume by shells
    Using the initial equations, solved for y, volume will be the volume of y=x^2 - volume of y=\sqrt{x}. from x=0 to x=1.

    So volume = 2\pi \int_0^1x(x^2)~dx - 2\pi \int_0^1 x(\sqrt{x})~dx

    V = 2\pi \int_0^1 (x^3-x^{3/2})~dx

    V = 2\pi (\frac 14x^4-\frac 25x^{5/2})_0^1

    V = 2\pi (\frac 14-\frac 25)

    V = 2\pi (-\frac 3{20})

    V = -\frac{6}{20}\pi






    So either one or both of my ways are wrong >.<
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by angel.white View Post
    Question: Let V be the volume of the solid obtained by rotating about the y-axis the region bounded by y=\sqrt{x} and y=x^2. Find V both by slicing and by cylindrical shells. In both cases, draw a diagram to explain your method.


    My work:
    -------------------
    a) Volume by slices.
    Since this is rotated about the y-axis, I solved in terms of x:
    y=\sqrt{x} ~~~~\Rightarrow ~~~~x=y^2
    y=x^2 ~~~~~\Rightarrow ~~~~x = \sqrt{y}

    Then I said the cross-sectional areal will be \pi x^2 since x will be the radius.

    So I plugged my equations into it:
    Area_1 = \pi y
    Area_2 = \pi y^2

    Since they intersect at the points (0,0) and (1,1) Volume will be from y=0 to y=1:
    V=\int_0^1 (Area_1-Area_2)~dy

    V=\pi \int_0^1 (y-y^2)~dy

    V=\pi (\frac 12y^2-\frac 13y^3)_0^1

    V=\pi (\frac 12-\frac 13)

    V=\frac 16\pi

    [snip]
    The formula is V = \pi \int_{y=a}^{y=b} x^2 \, dy.

    So:

    V_1 = \pi \int_{y=0}^{y=1} y \, dy.

    Note: y = x^2 \Rightarrow x^2 = y.


    V_2 = \pi \int_{y=0}^{y=1} y^4 \, dy.

    Note: y = \sqrt{x} \Rightarrow x = y^2 \Rightarrow x^2 = y^4.


    V = V_1 - V_2 = \frac{3\, \pi}{10} cubic units.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by angel.white View Post
    [snip]
    b) volume by shells
    Using the initial equations, solved for y, volume will be the volume of y=x^2 - volume of y=\sqrt{x}. from x=0 to x=1.

    Mr F says: Other way around! That is, volume of {\color{red}y=\sqrt{x}} - volume of {\color{red}y=x^2}.

    Note: From a graph of the region, it's clear that the height of the shells is {\color{red} \sqrt{x} - x^2} since {\color{red}y=\sqrt{x}} is above {\color{red}y=x^2} over the interval 0 < x < 1.


    So volume = 2\pi \int_0^1x(x^2)~dx - 2\pi \int_0^1 x(\sqrt{x})~dx

    V = 2\pi \int_0^1 (x^3-x^{3/2})~dx

    V = 2\pi (\frac 14x^4-\frac 25x^{5/2})_0^1

    V = 2\pi (\frac 14-\frac 25)

    V = 2\pi (-\frac 3{20})

    V = -\frac{6}{20}\pi
    [snip]
    Very close. The answer is \frac{6}{20} \, \pi = \frac{3 \, \pi}{10} cubic units, in agreement with the answer found using the slices method.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member angel.white's Avatar
    Joined
    Oct 2007
    Posts
    723
    Awards
    1
    Thank you, I appreciate all the help you've been providing.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Volume By Slices
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 9th 2009, 12:35 AM
  2. Volume Around y=r With Shells
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 26th 2009, 06:01 PM
  3. volume(shells method)
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 2nd 2009, 09:18 AM
  4. Volume Using Shells
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 21st 2008, 03:54 PM
  5. Volume by cylindrical shells
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 9th 2008, 02:42 PM

Search Tags


/mathhelpforum @mathhelpforum