ES-180 Center of Gravity and Centroids by Integration
    
    
where
    xel - x-coordinate of CG of element dW
    yel - y-coordinate of CG of element dW
and the centroid
    
    
If a body is not composed of standard sections, but its area is integrable, the method of integration may be used to find the CG.

    

where

    xel = x
    yel = y/2
    dA = y dx
The location and calculation for a vertical element are shown here...

Note: the centroid of a rectangle is in its center.


    
where
    
    yel = y
    dA = (a-x) dy
For a horizontal element, the location and calculations are shown...

    
where
    
    
    
If the centroid or CG is desired in polar coordinate for ease of integration the diagram and calculations are shown here...>



Note: The differential area of the sector is a triangular element...

and the CG of a triangle is one third of the altitude from the base or two thirds the altitude from the apex.


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