Engineering Statics

F-26


Determine by direct integration the moment of inertia of the shaded area with respect to the x axis.

A horizontal cut will be used for dA. y is given as a function of x from 
the equation of the curve.
k is found by substituting into the equation of the curve the fact that 
when x=a y=b.
By subsituting the value of k into the curve x can be found as a function 
of y, a and b.
Note that dA=(a-x)dy because of the width of the cut is the total distance 
a minus the value of x at that height.
The value of x is substituted into the integral.  Since integration 
will proceed in the y direction the limits go from 0 to b.
The integrand is simplified.

The integral is separated into two integrals and all constants are factored out of the integrals for ease of integration." >
The integration is performed.
Limits are substituted into the expressions.
The value for the moment of inertia about the x axis is found.