f(x)=90x^8(1-x)=90(x^8-x^9). ∫f(x)dx=90(x^9/9-x^10/10)=90x^9(1/9-x/10) between limits {m,n}.
When m=0 and n=1 the integral=1 which is the sum of all probabilities.
The graph of this PDF is very much weighted to the right, so we can expect low probability values for x≤0.5.
(a) ∫f(x)dx for [0,5]=90(0.5^9)(1/9-1/20)=0.010742 approx.
(b) ∫f(x)dx for [0,0.25]=90(0.25^9)(1/9-1/40)=2.956E-5 approx., so for [0.25,0.5] P is 0.010742-0.000030=0.010713 approx.