What is the correct centroid for a semicircle?

[lizzyb]

The centroid of a semicircle is noted at being (4r)/(3 pi) - http://en.wikipedia.org/wiki/List_of_centroids. However, when I did the work myself using the integral of y da over the area, I came up with (2 r)/(pi). I figured I was doing something wrong so sought out someone else's work and found this:

http://books.google.com/books?id=P7...&dq=analytically centroid semicircle&pg=PA164

Who is right?

 

[AUMathTutor]

The correct answer is 4r/3pi, if what you are after is the average y-coordinate. This can be calculated fairly easily by doing this:

(1/A) S[y*2sqrt(1-y^2)dy] from 0 to 1.

I think you are confusing a semicircular area with a semicircular arc. Both appear on the Wikipedia page with centroids. The excerpt you posted clearly refers to the case of a semicircular arc, not an area.