Observe the figure:

The shaded region can be divided in to three parts.

There are two triangles and one rectangle at the bottom. 

The centroid of the shaded region is the average of the three centroids.

Formula for the triangle centroid is .

Formula for the triangle area is .

Find the centroid and area of triangle on the right.

From the figure, vertices of the the triangle are .

.

.

Area:

.

 Find the centroid and area of triangle on the left.

From the figure, vertices of the the triangle are .

.

.

Area:

.

 From the figure:

The rectangle is symmetric with respect to the line and .

Centroid of the rectangle is .

Area of the rectangle is base times height.

.

The centroid of the shaded region is the average of the three centroids.

, and .

, , and .

.

Centroid is .