AIME I 2010 #11

[stupidityismygam]

Let [unparseable or potentially dangerous latex formula] be the region consisting of the set of points in the coordinate plane that satisfy both [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]. When [unparseable or potentially dangerous latex formula] is revolved around the line whose equation is [unparseable or potentially dangerous latex formula], the volume of the resulting solid is [unparseable or potentially dangerous latex formula], where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are positive integers, [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are relatively prime, and [unparseable or potentially dangerous latex formula] is not divisible by the square of any prime. Find [unparseable or potentially dangerous latex formula]

[88bobcat]

[stupidityismygam]

Non-Calculus Approach:

The unrotated region forms a triangle with endpoints [unparseable or potentially dangerous latex formula]. Draw the line parallel to [unparseable or potentially dangerous latex formula] through [unparseable or potentially dangerous latex formula] so the equation of that line is [unparseable or potentially dangerous latex formula]. Take perpendicular from [unparseable or potentially dangerous latex formula] to [unparseable or potentially dangerous latex formula] to form two triangles. The rotated image will be the sum of two cones.

The radius is the perpendicular distance between the two parallel lines (use [unparseable or potentially dangerous latex formula]) and the sum of their heights is the distance between [unparseable or potentially dangerous latex formula] which is [unparseable or potentially dangerous latex formula]

Thus, the desired volume is [unparseable or potentially dangerous latex formula]