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Math Help - volume of the solid of revolution

  1. #1
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    volume of the solid of revolution

    region R is bounded by the curve y=ln x, and the lines y=1 and x=1. Find the volume of the solid formed when R is rotated through 360 degree about the y-axis.

    What is the graph look like and how to integrate ?
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by tommylai12 View Post
    region R is bounded by the curve y=ln x, and the lines y=1 and x=1. Find the volume of the solid formed when R is rotated through 360 degree about the y-axis.

    What is the graph look like and how to integrate ?
    I am attaching the region in a pdf file.

    Use the washer method for this problem, or the shell method if you are familiar with both. Below I quickly discuss the shell method:

    V = INTEGRAL ( distance traveled * height of region)

    distance = 2 pi x since the radius from the rotational axis to a level x is simply the x coordinate

    height of region = 1 - ln x

    the bounds are from x = 1 to the intersection point between y=ln x and y=1, which is simply e = 2.71828...

    So the volume is INTEGRAL (2pi x (1-ln x)) from 1 to e.

    The washer method, briefly:

    Inner radius: r(y) = 1
    Outer radius R(y) = x = e^y
    Bounds y = 0 to y =1

    Formula: V = INTEGRAL (Pi R^2 - Pi r^2)

    Good luck!!
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  3. #3
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    Thanks for your help. It helps a lot
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  4. #4
    Senior Member apcalculus's Avatar
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    You're welcome. The answer turns out to be: 6.894313198.

    Attached find a 3D image of the solid. The horizontal line extending from left to right on the green surface is y = 1, and the curve to the right (concave down) is the graph of ln (x).
    Attached Thumbnails Attached Thumbnails volume of the solid of revolution-3dsolid.jpg  
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  5. #5
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    Quote Originally Posted by tommylai12 View Post
    region R is bounded by the curve y=ln x, and the lines y=1 and x=1. Find the volume of the solid formed when R is rotated through 360 degree about the y-axis.

    What is the graph look like and how to integrate ?
    Volume=Pi*int{(e^y)^2-1^1)}[/tex]

    The region:
    Attached Thumbnails Attached Thumbnails volume of the solid of revolution-region.gif  
    Last edited by curvature; April 29th 2009 at 08:33 AM.
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  6. #6
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    Quote Originally Posted by apcalculus View Post
    You're welcome. The answer turns out to be: 6.894313198.

    Attached find a 3D image of the solid. The horizontal line extending from left to right on the green surface is y = 1, and the curve to the right (concave down) is the graph of ln (x).

    So that mean integrate lnx from 0 to 1?

    I know already... INTEGRAL (2pi x (1-ln x)) from 1 to e...thanks.
    Last edited by mr fantastic; April 29th 2009 at 10:52 PM. Reason: Merged posts
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