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Calculus/Volume of a Rotation (Integration)

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Expert: - 10/15/2008


Question
An instructor told me if I am bored in calc 2 to try volume of rotation about
something with a slope... I attempted to calculate the volume of y=x^2
rotated about y=2x. I uh, ran into difficulties trying to set up an equation for
the radius. I figure if I could either find an equation for the radius or perhaps
an equation for this curve rotated up to the y-axis I could go ahead and use
the disk method, or shell method or whatnot, but I have run out of ideas. It
has been one day. lol any help would be awesome.

Answer
The endpoint are where x²=2x (there are two solutions).
The radius of rotation is y=x²-2x as long as we are still rotating each point in the y direction.  If we start to rotate perpendicular to the line y=2x, it gets a whole lot more complicated.

You would have to calculate the length of the line going from y=2x to the curve that had slope -1/2 (since perpendicular means -1/slope).

In the simple case, take the integral(0 to 2)(π(x²-2x)²)dx.
Square out the integral, move the π outside, and then itegrate that polynomial.

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