How to calculate the product moment of area Ixy for asymmetric plane PQRS

[phydis]

Here as shown in the diagram,

(1)Is Ixy zero for plane ABCD symmetrical along X axis? If yes, how it happens?
(2)How can we calculate Ixy for the plane PQRS ?

Ixy = the product of moment of area.

Thanks!

 

[SteamKing]

For any plane figure with an axis of symmetry, Ixy = 0 for coordinate axes thru the centroid of the figure.

To confirm this, examine the definition of Ixy = Int(x*y)dA For every element of area dA which lies on one side of the axis of symmetry, there will be a corresponding element of area lying on the opposite side which is the same distance from the axis. When you add up all of these elements of area multiplied by x*y, the integral is necessarily equal to zero.

To calculate the Ixy of rectangle PQRS, apply the parallel axis theorem.