Like the title says how do I use this. Especially in a case where $P(X \le 0.5)$ and harder cases where there are 2 bounds like $P(0.25 \lt X \lt 0.5)$. I am using a TI-84 graphic calculator
The question I tried to answer had X ~ Beta(3,2). Where the question was find $P( (1/3) \lt X \lt (1/2))$ My logic was to split it in 3 different probabilities namely: $(P(X \leq (1/2)) - P(X = (1/2)) - P(X \leq (1/3))$ Seeing as I can do the integral from 0 to 0.5 minus 0.5 to exclude it and then the whole thing minus the integral from 0 to 1/3. the answer I get is -0,216
It would be better for you to use R, SPSS or something like that to calculate that probabilities. I calculate it and get $0.2013889...$ I think your fault is calculating $P\{X=1/2\}$, because this probability is, of course, zero (beta distribution is continuous, not discrete). May be your calculator get it wrong in that calculation.
EDIT: But it is also quite easy to calculate directly $$\int_0^{1/2} \frac{t^2(1-t)}{B(3,2)}dt $$ where we have $1/B(3,2)=12.$ Something similar we have to do for the $P\{X<1/3\}.$ Or even more easier just calculate the integral: $$12\int_{1/3}^{1/2} (t^2(1-t))dt. $$
How I calculate $\frac{1}{B(3,2)}$? Like the link, that I give in a comment for Beta function, says I can calculate $B(3,2)$ like $\frac{\Gamma(3)\Gamma(2)}{\Gamma(5)}=\frac{2!1!}{4!}=\frac{2}{24}=\frac{1}{12}$, so $$\frac{1}{B(3,2)}=\frac{1}{1/12}=12 $$