Engineering Statics

F-6


Locate by direct integration the centroid of the area shown.

The first thing we do is draw a diagram of the shape and the slice we 
will use during integration.
The area dA is equation to the height (y<sub>1</sub>-y<sub>2</sub>) times 
the width, dx.
The upper line has the equation y = mx and the lower curve has the equation 
y=kx<sup>3</sup>.  <BR>These are substituted into the equation for the two 
values of y.
The constants m and k can be found from the values a and b given on the 
diagram.  <BR>The slope of the line, m, is b/a.  For the curve when y=b, 
x=a, hence the equation becomes b=ka<sup>3</sup>.
The values for m and k are substituted into the expression for dA.
Integrating with limits on x from 0 to a:
The integration is performed.
The upper and lower limits are substituted in.
Simplifying:
The centroid is defined as the first moment of the area divided by the area.
Notice that the integral is the integral for finding the area with each 
terms premultiplied by x.<BR>We are adding up all the first moments of the 
infinitessimal areas dA.<BR>  The term (4/ab) is the reciprocal of the area. 
<BR>In other words, we are dividing by the area to immediately get the centroid.
Simplification sets up the integral.
Integrating:
Substituting limits and cancelling:
Simplifying:
Simplifying:
Simplifying gives the location of the centroid in the x direction.
The definition of the centroid in the y-direction is given.
The same value for dA is substituted as was in the integral for the centroid 
in the x-direction.  <BR>Note the term (y<sub>1</sub>+y<sub>2</sub>)/2.<BR>This term 
is the distance from the x axis to the centroid of the area dA.<BR>This is the average 
of the distance to the top and the distance to the bottom of the area dA.
The expressions arae simplified.
Values for y<sub>1</sub> and y<sub>2</sub> are substituted.  <BR>The term 
2/ab is the reciprocal of the area A, divided by 2.  <BR>The 2 appears from 
the denominator for the average of the vertical distances.
The expression is integrated.
Limits of integration are substituted.
Simplifying:
Simplifying:
The centroid in the y direction is found.