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Math Help - Volume of region bounded by two graphs?

  1. #1
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    Volume of region bounded by two graphs?

    The region is bounded by y=x^3 and y=x in the first quadrant.

    I already found the area, but I'm having trouble with the volume.

    First:
    A)find the volume rotated around y=2
    B) find volume rotated around the y axis
    C)the area (1/4) is the base of a solid. There are cross sections perpendicular to the x-axis that are semi circles. Find the volume of the solid.

    Help?
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  2. #2
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    The formula to find the volume is:

    <br />
\pi \int (outer - line going around)^2 - (inner - line going around)^2<br />

    That means "a" would be setup as follows:

    <br />
2 \times \pi \int (x-2)^2 - (x^3-2)^2<br />

    The integral will be from 0 to 1.

    The reason we multiply it by 2 is because the two lines actually form a solid twice. Therefore we will take the integral from a point of intersection to the next point of intersection on one solid and multiply times 2 in order to get the volume of both solids.

    b)To find the area going around the y-axis we need to place all of our x's in terms of y.

    We come out with x=\sqrt[3]{y} and x=y. Remember the graph looks the same, but the equations are just in terms of y.

    Then just use the formula to find the volume.

    Your setup should look as follows:

    2 \times \pi \int (\sqrt[3]{y})^2 - (y)^2

    The integral will be from 0 to 1 again because the graphs intersect at those points.

    Question "c" is beyond my knowledge. Maybe someone else here can enlighten me.

    If you have any questions, just ask!
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  3. #3
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    So, would the graph be considered dx or dy? Because oringally I thought it was dx, but then it asked for it to be rotated around y=2 and the y-axis. So, in order to do that, don't I need to make the y=2 volume change from x to y like in (b)?
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  4. #4
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    No, for "a" your terms need to be in terms of x.

    This is because y=2 is virtually the same as the x-axis, but shifted up 2.
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  5. #5
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    ok. And it's the washer method because of the formula used? Btw, I appreciate your help!
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  6. #6
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    You got it!

    No problem and good luck!
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  7. #7
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    Ok. That's probably what I was doing wrong. I had thought it was the shell method because it made a "cup" shape. Haha.
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