(9/20, 9/20) is correct. The region is obviously symmetric about the line y = x, but it's not symmetric about the perpendicular line x+y = 1. In fact, it is a lens-shaped region that is fatter towards the (0,0)-end than towards the (1,1)-end.
I just got back from my calculus midterm, and there was one question that I'm not sure if I got correct...
What is the center of the region bound by y=x^2 and y=sqrt_x on 0<x<1
When I did the math I came up with (9/20, 9/20), but intuitively I thought it would be (1/2, 1/2).
I tried searching google because it seems like a cookie-cutter question, but didn't have much luck. I'm just wondering if I got it right.