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Math Help - Volumes of Revolution

  1. #1
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    Unhappy Volumes of Revolution

    find the volume of the area enclosed by Y=X^2+1, Y=0, X=2 and revolved around X=3
    Using the Disk Method

    I used the shell method and got: 175pi/3

    But I need to know how to set up the disk integral

    Also,

    Y=X^2+1, X=0, Y=5 revolved around Y=7

    using the shell method i think i got 12pi, but that may be wrong, and I need to know how to set up the disk method.

    thanks so much
    Last edited by terrytriangle; February 25th 2009 at 03:34 PM.
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  2. #2
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    Volume of revolution

    Hello terrytriangle
    Quote Originally Posted by terrytriangle View Post
    find the volume of the area enclosed by Y=X^2+1, Y=0, X=2 and revolved around X=3
    Please clarify this question. The curve y = x^2 +1, the line y =0 (i.e. the x-axis) and the line x = 2 does not enclose an area. Do you mean the area enclosed by

    • y=x^2 +1, y=0, x=2 and x=3?
    • Or y=x^2+1, x=0, y=0 and x=2?
    • Or something else?

    Grandad
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  3. #3
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    Quote Originally Posted by terrytriangle View Post
    find the volume of the area enclosed by Y=X^2+1, Y=0, X=2 and revolved around X=3
    Using the Disk Method

    I used the shell method and got: 175pi/3

    ...
    1. If I understand your question correctly the solid looks like a kind of funnel (upside down). (See attachment)

    2. The solid consists of two different kinds of solids:
    The base is a cylindrical ring with height of 1.
    The top is the funnel mentioned above.

    3. The base area of one disk is

    b = \pi((3-x)^2-1^2)=(x^2-6x+8)\cdot \pi

    the height of one disk is

    h = dy From

    \dfrac{dy}{dx} = 2x~\implies~dy=2x\cdot dx

    4. The complete volume is:

    V = \pi(3^2-1^2)\cdot 1+\int_0^2\left(\pi((x-3)^2-1^2)\cdot 2x  \right)dx

    I leave the rest for you.


    By the way: Your result must be too large: The solid can be placed completely into a cylinder with a base diameter of 6 and a height of 5. Such a cylinder has a volume of 45\pi
    Attached Thumbnails Attached Thumbnails Volumes of Revolution-rot_trichter.png  
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  4. #4
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    thank you i figured it out now
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  5. #5
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    earboth, terrytriangle said, "find the volume of the area enclosed by Y=X^2+1, Y=0, X=2 and revolved around X=3", which, as Granddad said, doesn't quite make sense. You have assumed the region enclosed by y= x^2+ 1, y= 0, x= 2, x= 3, revolved around x= 0.
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  6. #6
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    Quote Originally Posted by HallsofIvy View Post
    earboth, terrytriangle said, "find the volume of the area enclosed by Y=X^2+1, Y=0, X=2 and revolved around X=3", which, as Granddad said, doesn't quite make sense. You have assumed the region enclosed by y= x^2+ 1, y= 0, x= 2, x= 3, revolved around x= 0.
    The only additional assumption I made was:

    "find the volume of the area enclosed by Y=X^2+1, Y=0, X=2, x = 0 and revolved around X=3"
    Attached Thumbnails Attached Thumbnails Volumes of Revolution-rot_trichter.png  
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