Thank you arildno,
I did that integral sometime in the past 3 years and was racking my memory to come up with it; I found another nonsense solution that agreed with the known answer but as in the one I posted (after 2 days and maybe 6 hours surfing the web searching for answers), the limits of integration didn't make sense.
Even so , the polar coordinate translations are basic and I am chagrined. It was not a waste however as this is the first time I have heard the term area moment of inertia even though I have done a lot of mass moments and radii of gyration calculations. The EulerBernoulli Beam Equation was under review here and it is fascinating:
[tex]\frac{d^2}{dx^2}\left[E I \frac{d^2w}{dx^2}\right]=\rho[/tex] where E is Young's Modulus, I is the area moment of inertia, w is the out of plane displacement and [tex]\rho[/tex] is force acting downward on a very short segment and has units of Force per unit length (distributed loading). The xaxis is the lengthwise polar axis passing through the center of the beam. If E and I do not vary with x, then [tex] E I \frac{d^4w}{dx^4} = \rho [/tex] This is the first ODE I have come across that utilizes the fourth derivative and since the boundary conditions, depending on how the beam is supported include up to the third derivative, I wanted to make sure that I understood what the equation was saying and be able to relate to the area moment integral was critical. Thank you again.
