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How can I find the volume of an object when I am given the curves $y = x-x^2$ and $y= 0$, when it is rotated about the line $x= 2$. I understand that my height is $y = x-x^2$, and I want to say my radius is $x$, but I don't think that is right.

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up vote 1 down vote accepted

You’re right: it isn’t right. :-)

The radius is always the distance between the axis of revolution and the shell. For a shell at $x$, that distance is $2-x$ when $x\le 2$ and $x-2$ when $x\ge 2$. I expect that you’ve already discovered that the shells for your region range from $x=0$ to $x=1$, so the radius is always $2-x$ in this problem.

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Thank you very much! – Kyle H Dec 13 '12 at 4:01
@Kyle: You’re welcome! – Brian M. Scott Dec 13 '12 at 4:05

The picture is key here, so make sure you are working with an accurate one:

enter image description here

Since the axis of revolution is $x=2$, the radius of a shell emanates from there (not the $x$ axis), and so it is $2-x$.

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Thanks! I had a feeling the $2$ would be incorporated somehow. – Kyle H Dec 13 '12 at 4:02

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