The center of mass of the plate (or the centroid of ) is located at the point , where

and .

Theorem of Pappus :

Let be a plane region that lies entirely on one side of a line in the plane. If is rotated about , then the volume of the resulting solid is the product of the area of and the distance traveled by the centroid of .

i.e, .

A rectangle with sides and is divided into two parts and by an arc of a parabola that has its vertex at one corner of and passes through the opposite corner.

First find an equation of parabola that has its vertex at the origin and passes through the point  .

The standard form of parabola that has its vertex at the origin is .

Substitute the point in .

.

Thus, the parabola equation is .

 Find the area of two regions.

Let the area of two regions are and .

Area of the rectangle is .

Find the -coordinate of the centroid for .

The parabola is the top function.

Consider .

Find the -coordinate of the centroid for .

The parabola is the top function.

Consider .

Therefore, the centroid of is .

Find the -coordinate of the centroid for .

The parabola is the top function.

Consider .

Find the -coordinate of the centroid for .

The parabola is the top function.

Consider .

Therefore, the centroid of is .

Centroids of and are and .