I am trying to generate a function that represents essentially a Beta Distribution that revolved around the Y axis. To do this I am trying to integrate only one quadrant because the expansion from that is trivial. To do this I am using the following code:
$Assumptions=0<x<1 && \[Alpha]>1 && \[Beta] >1&&x<R<1;
dy=D[Sqrt[R^2-x^2],R]
CurveFunc=dy*R^(\[Alpha]-1)*(1-R)^(\[Beta]-1)/Beta[\[Alpha],\[Beta]];
CurvePDF=Expand[Integrate[CurveFunc,{R,x,1} ]]
This produces the following Integral with primarily made up with hypergeometric functions:
But a quick review of that function and you will see that it is undefined or singular at Integer values of [Alpha]. Which is incorrect because if you substitute in an integer value of [Alpha] before hand it will produce a valid return ie:
CurvePDF=Expand[Integrate[CurveFunc/.\[Alpha]->6,{R,x,1} ]]
Will return a function that is continuous. I would like to find the generic integral that works for all real Alpha > 1. Does anyone have an idea how to get Mathematica to produce an integral without those singularities?