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 P: 13 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 Euler-Bernoulli Beam Equation was under review here and it is fascinating: $$\frac{d^2}{dx^2}\left[E I \frac{d^2w}{dx^2}\right]=\rho$$ where E is Young's Modulus, I is the area moment of inertia, w is the out of plane displacement and $$\rho$$ is force acting downward on a very short segment and has units of Force per unit length (distributed loading). The x-axis is the lengthwise polar axis passing through the center of the beam. If E and I do not vary with x, then $$E I \frac{d^4w}{dx^4} = \rho$$ 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.