Calculating Ixy | Plane ABCD, PQRS Symmetry & Moment of Area

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For a plane figure symmetrical along the X-axis, the product of moment of area Ixy is zero when calculated through the centroid. This occurs because each area element on one side of the axis has a corresponding element on the opposite side, resulting in equal and opposite contributions to the integral. To compute Ixy for a rectangle like PQRS, the parallel axis theorem can be utilized. This theorem allows for the calculation of moments of area by shifting the axis to the centroid. Understanding these principles is crucial for accurate moment of area calculations in structural analysis.
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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!
 

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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.
 
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