This alternate form of input
int(x^2*y^2*sqrt(R^2-y^2-x^2),{x,-sqrt(R^2-y^2),sqrt(R^2-y^2)})
is understood by WolframAlpha, but it does not find an exact symbolic solution for that.
WolframAlpha is able to do the indefinite integral
int(x^2*y^2*sqrt(R^2-y^2-x^2),{x})
giving
(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
Integrate[x^2*y^2*Sqrt[R^2-y^2-x^2],{x,-Sqrt[R^2-y^2],Sqrt[R^2-y^2]}]
gives
(Pi*(-(R^2*y) + y^3)^2)/8