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I just thought about this idea and I decided to work on it.

After taking on a general case, which proved to be too difficult, I tried a specific case. Something simple like the curve $y_1 = x^2$ rotating about the line $y = x$

Which is the same as rotating $y = \sqrt{x}$ about the x-axis.

I know I need to find the new radius which is the line perpendicular to y = x and I need to pick a particular point on the curve and the line.

So if i were to pick say, x = 0.5, the perpendicular line would be

$y =-x + 1$

So my solid of revolution integration would be

$\pi \int_{a}^{b} (-x + 1)^2 d?$

Unfortunately it proved to be very difficult to find the slanted differential in terms of dx and I couldn't figure out what the change of variables of bounds were.

Any ideas?

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    One thing you could do is to rotate your coordinate system such that the once oblique lines are parallel to the coordinate axes. It helps if you first express the curve you're rotating about the axis in parametric form...2012-02-02
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    But that's avoiding the problem...2012-02-02
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    There's really no point in persisting in one coordinate system when there's another that makes manipulations vastly easier.2012-02-02

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