Are you clear that D is just a circular cone with a = 1 and h = 1? Cylindrical...

z is going...

from z = r (the side of the cone) up to z = 1 (the base)

... for every bit that r is going...

from zero (the z axis) along to as far as where r = z = 1

... for every bit that theta is going right round (zero to 2 pi).

And your integrand is r(x^2 + y^2) = r^3.