\fFor the next problems, let M be the portion of the paraboloid z = x2 + :92 in the rst octant bounded by the cylinder 1:2 + y2 = 3. the plane 1; = a: and the zzplane. More precisely, M={(m,y,z) :z=m2+y2,wz+y2 3,03y5x}. Here's a picture (just so you won't say I never gave you any picture): (Notice that this sketch is correct because we wrote an M nmtt to the surface, so it must really be M} (3) Using cylindrical coordinates, nd a parametrization @(r, 9) for M, clearly indicating its domain. Compute the normal vector '1',- X 'I>.9, and decide if it is upward (positive zcomponent) or downward (negative zcomponent). [Hint to find the bounds forr and 6', use the inequalities in the description of Mt Don't worn,t so much about the picture] (b) Consider a fluid with density 1 1(2, y, z) = 1+x2 + 32 measured in g/in' flowing with velocity v(x, y, z) = (x, y, 1) measured in in/ s. The net amount of fluid that goes through M per unit of time (in this problem, per second) in a specified direction (one of two possible orientations for M) is referred to as the mass flow rate & of the fluid through M. It is given by the surface integral 1 HO . Nas , where / denotes a normal field to M compatible with the orientation we are interested in. Show that the net amount of fluid that passes through our surface M going down after 4/3 seconds is 3 M Mv . NaS = (1 -log(2)) T grams. Here, "going down" means that we choose an orientation for M so N is a downward normal vector. (c) Notice that the quantity you found in (b) is positive. From a physical perspective, what is the meaning of the positive sign in this computation? Use a single sentence to answer this