Engineering Statics

F-28


Determine the moment of inertia and radius of gyration of the shaded area with respect to the y axis.

The equations for both curves are given.
The value of k<sub>1</sub> is found by substituting into the equation of 
the curve the fact that when x=a y=b.
The value of k<sub>2</sub> is found by substituting into the equation of 
the curve the fact that when x=a y=b.
A vertical cut is taken to set up the expression for dA.  Note that 
the lower curve's value of y must be subtracted from the upper curve's 
value of y to obtained the height between the two curves.
The substitution that y=f(x) is made for both curves.
The expressions are substituted into the integral.  Since the integration 
is in the x direction the limits go from 0 to a.

The integration is performed.
Limits of integration are substituted into the expressions.
Terms are simplified.
The moment of inertia about the y axis is found.
The area between the two curves is found again by using vertical cuts and 
the same substitution for dA as used for the moment of inertia calculation.
The integral is separated into two integrals.
The integration is performed and limits are substituted.
The terms are simplified.

The area is found.
The expression for the radius of gyration is given.
The expressions for the moment of inertia and area are substituted into 
the equation for the radius of gyration.
Expression is simplified and the radius of gyration found.