Originally Posted by Crossplot
I have been doing some crude "hand cranking" and found that "r/2' works all over the cylinder.
Multiplying the radial pressure gradient by "r/2" amounts to "bringing an algebra weapon to a calculus fight". Your approach works in this specific case, but only because multiplication by "r/2" transforms the expression for the radial pressure gradient into an expression for the dynamic pressure. For steady, irrotational, uniform-density flow, the increase in dynamic pressure from one location to another is equal to the decrease in (static) pressure.
Just because an analysis leads to an equation that accurately predicts the relationship between flow parameters (in this case the Bernoulli equation), that doesn't mean the analysis is correct. The are an unlimited number of ways you can arrive at the Bernoulli equation (monkeys with typewriters comes to mind), however, the reason the Bernoulli equation accurately predicts the relationship between velocity and pressure in a steady, irrotational, uniform-density flow is because it can be derived from conservation of momentum.
I have updated the .pdf in the post above to address the issues associated with analyzing the flow around an airfoil (or cylinder) from a reference frame where the airfoil is moving through otherwise still air.