Does this match what you expect?
Integrate[Integrate[y^2,{x,xlow(y),xhigh(y)}],{y,ylow,yhigh}]
so
Integrate[Integrate[y^2,{x,y^2,2+y}],{y,-1,2}]
Giving exactly that to either WolframAlpha or to Mathematica returns 63/20==3.15
It is also possible to do the same with
Integrate[y^2,{y,-1,2},{x,y^2,2+y}]
but the y and x must be given in exactly that order if you use that form.
As a check, If you are using Mathematica or other calculation tools then it is possible to calculate a crude manual approximation of this
total=0;d=1/100;
For[y=-1,y<=2,y=y+d,
For[x=y^2,x<=2+y,x=x+d,
total=total+y^2*d^2
]];
total//N
which returns
3.1652
and that is approximately the same as the result from the integration and smaller values for d will give better approximations..
But after doing that be sure to tell Mathematica to forget the cached assigned values to x and y with
x=.;y=.;
before doing further Integrate which do not expect to have x or y have been assigned values.
