Your triple integration is
Integrate[xy*sqrt(x^2+y^2+z^2),Element[{x,y,z},
ImplicitRegion[z>=sqrt(x^2+y^2)&&x^2+y^2+z^2<=1,{x,y,z}]]]
While this might not get you an answer, you have some syntax errors that need fixing. You need to use
Sqrt[...]
rather than
sqrt(...)
And xy
needs a space between the x
and y
.
If we Reduce
the restrictions, we can end up with the limits of integration:
Reduce[{z > Sqrt[x^2 + y^2], x^2 + y^2 + z^2 < 1}]
(* (0 < z <= 1/Sqrt[2] && -Sqrt[z^2] < y < Sqrt[z^2] &&
-Sqrt[-y^2 + z^2] < x < Sqrt[-y^2 + z^2]) ||
(1/Sqrt[2] < z < 1 && -Sqrt[1 - z^2] < y < Sqrt[1 - z^2] &&
-Sqrt[1 - y^2 - z^2] < x < Sqrt[1 - y^2 - z^2]) *)
And the integral is expressed as follows:
Integrate[x y*Sqrt[x^2 + y^2 + z^2], {z, 0, 1/Sqrt[2]}, {y, -z, z},
{x, -Sqrt[z^2 - y^2], Sqrt[z^2 - y^2]}] +
Integrate[x y*Sqrt[x^2 + y^2 + z^2], {z, 1/Sqrt[2], 1}, {y, -Sqrt[1 - z^2], Sqrt[1 - z^2]},
{x, -Sqrt[1 - y^2 - z^2], Sqrt[1 - y^2 - z^2]}]
(* 0 + 0 = 0 *)