Find the exact coordinates of the centroid for the region bounded by the curves y=x, y=1/x, y=0, and x=2

Refer to the figure shown below.
The shaded area is common to the given curves.

The curves y=x and y = 1/x intersect at (1,1).
The area of the shaded region is


If the centroid of the shaded area is (x₁, y₁), then

Also,
The curve y = 1/x intersects x=2 at y = 1/2.
Therefore


The centroid is located at (1.6524, 1.1361)