find centroid of the region in the first quadrant bounded by given curves: y= x^7 , x=y^7

Answer 1

I think that the region boundaries must be augmented by constraints such as

x < M ; y < M for some positive real number M

In that case the x coordinate of the centroid is:

Cx = ∫ x x^7 dx / ∫ x x^7 dx with limits 0 and M

This is easily evaluated with the result Cx = (8/9 ) M

and by symmetry Cy = (8/9) M