let M be the portion of the paraboloid=²+² in the first octant bounded by the cylinder²+²=3, the plane=and the-plane. More precisely,

={(,,):=²+², ²+²3, 0}.

i) Using cylindrical coordinates, find a parametrization (,) for, clearly indicating its domain. Compute the normal vector_r, and decide if it is upward (positive-component) or downward (negative-component).

ii)Consider a fluid with density(,,) =1 / (1+x²+y²) measured in/³ flowing with velocity

(,,)=(, , 1) measured in/.The net amount of fluid that goes throughper second of time in a specified direction is referred to as themass flow rateof the fluid through. It is given bythesurface integral:

. N dS. M