I have these questions " I already have the answers"

I just need to know why I have to use divergence theorem in question 1 , why not stokes' for example? and for the second why it's just a surface integral? and for each problem! I need to know how can I figure out which method should I use for each problem!!

thank you in advance

I the 5 770 6 we e a S the the Evaluate Fods where F cxigies and s is the surface arround the slid region x + y + 2 s 4 and Gainen i Sketch to we can wire divergence F - and Fonds S; ********* want SF.ds S Calculate Fods Foxzi+x} + yk , s is the hemisphere x + 9 + 2-25 y of onented towards Gwen that shkes Fcxz ix, y> IS Fods Sfonds s S hemsphere Xety +2 25 32- Evaluate SF-ds s is the hemisphere x + y + 229, forgeurt, 2.70 F- (2y cast, et sine rets F - Lagoz, et senz, xes Se hemophere "4 ****, 12 want elivergente S SS Fods $ 3 Wat. Compute J Fids where F. is is the part of the sphere x + y2 + 2 - 25 that he above the plane 2-4 Stokes' Giron Foxz, yz, xy> S: is the part of the sphere *+9+27=25 above z. 4 o IF lr where r = xi + y) + z R is consists of the hemisfere Div Z=11-xky and the disk x+y's 1 in the xy-plane 2 Z+x+y - 1