Take the line segment whose endpoints are the points (1,0) and (-1,0) and a point of rotation (x,y).

If the segment is rotated all the way around the point it will trace out an annulus.

Find simplified formula for the area of this annulus in terms of x and y.

Redo this problem for a rectangle with corners at (1,1), (-1,1), (-1,-1), and (1,-1).

(In reply to part 1 not finished. by Jer)

You are right. The solution is for the wrong problem. In the original formula you quote, or the corrected one,

|pi*(4*x + 2*y^2)|,

it's true that for the origin, the formula gives zero.

The formula is for an annulus that's between the circles formed by the endpoints. That should be the case only if the perpendicular from the center of rotation to the line containing the given segment does not fall within that segment. If however the perpendicular does meet the line within the given segment, the annulus should be only between the farther endpoint of the segment and the point on the segment where the perpendicular falls.  I haven't done that calculation or see how to make that a simple formula without and if...then...else.

Posted by Charlie on 2010-12-21 11:34:39