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Math Help - Calculate volume with respect to y

  1. #1
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    Post Calculate volume with respect to y

    y=\ln{x} , 1\leq x\leq e

    Calculate the volume between y and the x-axis with:

    a) integration with respect to x.

    b)integration with respect to y.

    I did a) without any problem, but I don't get how to do with b).

    The correct answer are 2\pi \int_{0}^{1} y(e-e^y) dy

    How should I think? I get that I should change to equation from y= \ln{x} , 1\leq x\leq e to x=e^y , 0\leq y\leq 1 and that the but then I got no idea at all...
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  2. #2
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    Use another formula to find volume :
     V = 2\pi \int (y-k) x dy , k is the axis of rotation

    Because the integral is (dy), the part that are being integrated is between the curve and y-axis.
    First, you find the volume of the section that are bounded by y-axis and x = e, then find the volume of the section that are bounded by the curve and y-axis.

    Find out which one has bigger volume then substract it with the smaller one.

    Sorry, it's difficult for me to explain it with words....
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  3. #3
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    Quote Originally Posted by kallekall View Post
    y=\ln{x} , 1\leq x\leq e

    Calculate the volume between y and the x-axis with:
    Volume between y and the x-axis? Are you assuming rotation around the x-axis? It would be nice to tell us!

    a) integration with respect to x.

    b)integration with respect to y.

    I did a) without any problem, but I don't get how to do with b).

    The correct answer are 2\pi \int_{0}^{1} y(e-e^y) dy

    How should I think? I get that I should change to equation from y= \ln{x} , 1\leq x\leq e to x=e^y , 0\leq y\leq 1 and that the but then I got no idea at all...
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  4. #4
    Senior Member DeMath's Avatar
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    Also look carefully this picture

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  5. #5
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    Quote Originally Posted by HallsofIvy View Post
    Volume between y and the x-axis? Are you assuming rotation around the x-axis? It would be nice to tell us!
    Sorry songoku and HallsofIvy, I was a bit unclear there... I'm suppose to rotate around the x-axis.

    Later found this in the tips-section:

    ("Lösning" means solution, if anyone were wondering )
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  6. #6
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    1. Find the volume when area bounded by x = e and y-axis is rotated around the x-axis
    2. Find the volume when area bounded by the curve and y-axis is rotated around the x-axis

    Use formula : <br />
V = 2\pi \int (y-k)   x      dy<br />
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