#1 ... method of cylindrical shells

#2 ... Pappus's Centroid Theorem -- from Wolfram MathWorld

2nd Theorem of Pappus (a much easier method in this case) says ...

using cylindrical shells ...

let the small circle of radius be centered at the origin, with rotation radius .

first integral is just the area of a semicircle of radius multiplied by the outside constant ...

using a substitution on the second integral ...

... note that the limits of integration both become 0 upon substitution.

... which agrees w/ the calculation using Pappus