how do i derive equation of area,moment of inertia with respect to x axis for non uniform cantilever beam?
For this post, consider the x-axis as being along the beam's length, the y-axis as being up, and the z-axis as sticking out from the side of the beam.
You just need to create the function you're looking for. Let's take a sample cantilever beam that is 10' long, is rectangular in cross section, and the depth varies from 14" deep at the fixed point to 8" deep at the free end and is 8" wide. We know that the standard equation for the moment of inertia for a rectangle is zy^3/12. However, in our case the y varies. Assuming that x=0 at the fixed end, y = 14 - .05*x, where x is measured in inches. Your moment of area equation for this beam is simply the basic formula replacing y with the varying depth expression:
I = (8(14-0.5x)^3)/12
If your beam has a varying width, you just replace the "8" with the expression for the width in terms of x.