The curves of ##y=x## and ##y=sin^{-1} x## start with the second below the first, then intersect so the second is above the first, then meet again at ##x=1##.
You can get a lower (upper) bound by integrating the square of the lower (higher) of the two from 1/2 up to the first intersection point, then doing the same again from that point up to 1 (noting that the lower and upper will have switched at the first intersection).
Wolfram tells me that ##(sin^{-1} x)^2## has a closed form antiderivative, so you should be able to obtain all derivatives.
That will give you upper and lower bounds, but I don't know if they are narrow enough to meet the problem spec. Worth a try anyway.
EDIT: Ignore this. Vela's approach is much easier. The above gives much tighter bounds but that is not required by the question. I was a little concerned about the degree of difficulty in this solution.