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Math Help - Cylindrical shells:

  1. #1
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    Cylindrical shells:

    The region bounded by the curves y = x^2 and y^2 = x is revolved about the line x = -2:
    a.)Find the rectangular elements parallel to the axis of revolution.

    2\pi\int_{0}^{1} (\sqrt{x} - x^2)(x - (-2))dx

    b.)Find the rectangular elements perpendicular to the axis of revolution.

    2\pi\int_{0}^{1} (y^2 - \sqrt{y} - (-2))(y)dy

    can you check if its correct....
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    The region bounded by the curves y = x^2 and y^2 = x is revolved about the line x = -2:
    a.)Find the rectangular elements parallel to the axis of revolution.

    2\pi\int_{0}^{1} (\sqrt{x} - x^2)({ \color {red} x - (-2)})dx
    what is in red should be 2 + x, do you see why?

    b.)Find the rectangular elements perpendicular to the axis of revolution.

    2\pi\int_{0}^{1} (y^2 - \sqrt{y} - (-2))(y)dy

    can you check if its correct....
    what do you mean exactly by finding the rectangular elements? it seems you just tried to find the volume by changing the functions of x into functions of y. if that is what you are trying to do, then you would have to use the disk method
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    disk method for part b
    \pi \int_{0}^{1} (2 + \sqrt{y} - y^2)^2 dy
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    disk method for part b
    \pi \int_{0}^{1} (2 + \sqrt{y} - y^2)^2 dy
    the inner radius is: 2 + y^2

    the outer radius is: 2 + \sqrt {y}

    the volume is: V = \pi \int_{0}^{1} \left[ ( \mbox {outer radius} )^2 - ( \mbox {inner radius} )^2 \right]~dy


    (did you draw a diagram? it helps)
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  5. #5
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    Hi again...
    is there really no way that shell method can be applied in part b?
    this lesson is about Cylindrical shells btw
    thx
    Last edited by ^_^Engineer_Adam^_^; September 24th 2007 at 04:05 AM.
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    Hi again...
    is there really no way that shell method can be applied?
    this lesson is about Cylindrical shells btw
    thx
    we already did it using shells... 2\pi\int_{0}^{1} (\sqrt{x} - x^2)(2 + x)dx

    but the second question asked for something different. i'm not exactly sure myself what it is asking for
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    Hi again...
    is there really no way that shell method can be applied in part b?
    this lesson is about Cylindrical shells btw
    thx
    If the dV---I think your "rectangular element" here---is parallel to the axis of rotation, the shell method is used.

    If the dV is perpendicular to the axis of rotation, then either disc or washer method are used, depending on whether the dV is attached to the axis of rotation or not.
    So shell method cannot be used in part (b).
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    Quote Originally Posted by ticbol View Post
    If the dV---I think your "rectangular element" here---is parallel to the axis of rotation, the shell method is used.

    If the dV is perpendicular to the axis of rotation, then either disc or washer method are used, depending on whether the dV is attached to the axis of rotation or not.
    So shell method cannot be used in part (b).
    here is how i remember when to use the Shell or the disk method.

    The D in Disk reminds me of different. that is, the variable that describes the axis of rotation is different from the variable i'm integrating with respect to. So i use the disk method if i am revolving a function of x about a line y = constant, or i am revolving a function of y about a line x = constant.

    The S in Shell reminds me of same. that is, if the variable that describes the axis of rotation is the same as the variable i am integrating with respect to, i use shell. so to revolve a function of x about a line x = constant, or a function of y about a line y = constant, i use the shell method.
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