I just have easy input:
But Wolfram don't understand it and don't want to calculate the result. I mean it is okay that we have some variables (R,y), but they are not variables, they are constants... And it should be cleaner to calculate native integral and simplify it. But Wolfram don't want it...
This alternate form of input
is understood by WolframAlpha, but it does not find an exact symbolic solution for that.
WolframAlpha is able to do the indefinite integral
(y^2 (x Sqrt[R^2-x^2-y^2](-R^2+2 x^2+y^2)+(R^2-y^2)^2 ArcTan[x/Sqrt[R^2-x^2-y^2]]))/8
and you might consider whether you could substitute in your upper and lower bounds for x and subtract those two to give your definite integral.
If you can provide the values for the constants then WolframAlpha might be able to directly provide your solution.
If it is of any help to you, the full version of Mathematica is able to find the definite integral
(Pi*(-(R^2*y) + y^3)^2)/8