Engineering Statics

F-7


Determine by direct integration the y coordinate of the centroid of the area shown.

A vertical slice has been taken to get dA = ydx.  The expression for y 
as a function of x is found from the equation of the curve defining the area.
The slices are integrated with the constant, h, factored out of the 
integral sign.  The limits of integration are from 0 to L.
The integral is broken into three integrals to find the area A.
The integration is performed.
Limits of integration are substituted into the expression and the 
expression is simplified.
The area of found in terms of h and L.
The definition of the y coordinate of the centroid is given. <BR>Note that 
for a vertical slice dA=ydx while for a horizontal slice dA=xdy.  <BR>If ydx 
is used the y represents the distance from the x axis to the centroid of 
the area dA.  <BR>Hence it is written as y<sub>el</sub>.  <BR>Since the element 
is a rectangle, the centroid of the element is in the middle hence equal 
to y/2.
Substitution for the location of the centroid of the element is made.
Simplifying:
The substitution y=f(x) is made and limits of integration are set.
The expression in the integral has been squared so that integration can 
proceed.
The integration is performed.
Limits of integration are substituted.
Like terms are added and the earlier found expression for the area is 
given as its reciprocal, 1/A.
Simplification yields the y coordinate of the centroid of the given area.