I am trying to compute the mass of the beta distribution between two points. This involves computing the following integral
$$I(x,y;\alpha,\beta) = \frac{1}{B(\alpha,\beta)}\int_{x}^yp^{\alpha-1}(1-p)^{\beta-1}dp$$
Using the CDF, the above is equivalent to
$$I(x,y;\alpha,\beta) = I_y(\alpha,\beta) - I_x(\alpha,\beta)$$
where $I_x()$ is the regularized incomplete beta function. Does the above expression simplify further? Are there good quality approximations?