Example 2: Locate the centroid ( and ) of the shaded area with respect to the reference axes.

Solution: To solve for the centroid location, we must first choose an appropriate differential area. In this example, we choose a horizontal rectangle of width xand height dy.

Differential area: dA = x dy
Moment arms of the differential area: (x/2,y)

The x coordinate of the centroid is found as

Notice that the limits of the integral are determined based on the domain of dy.

To find the y coordinate of the centroid, we use the same differential element and write

In this case, it is easier to modify the integral in the numerator by expressing dyas a funtion of x,changing the limits of the integral and then performing the integration as

Therefore, the centroid of the shaded area is at

Alternate Solution: Let's consider the solution with a different choice of differential element.

Differential area: dA = dx dy
Moment arms of the differential area: (x,y) Notice that the location of the differential area is at some arbitrary point in the shaded area.
 Because of the choice of differential area, the integral over the area changes into a double integral over x and y. In this case, the selection of limits must consider the boundaries in x and y directions.

The solution for the y coordinate of centroid can be obtained in a similar fashion.