Question

1. (6 marks) Find the volume of the solid enclosed by the paraboloid 2 = 1 - 22 - y2 and the coordinate planes of the first octant O = {(x, y, z) | x > 0, y > 0, z>0}. 2. (7 marks) Calculate SS/ (82 +93) dr dy dz. where E is the upper hemisphere x2 + y2 + 22 < 1 and 2 > 0. 3. (7 marks) Evaluate the integral SL (x + y) er?-y dA by making the appropriate change of variables where Rxy is the rectangle enclosed by the lines 1-y=0,2-y= 2, 3 + y = 0 and x +y = 3.

Answer 1

Z 7 16,011) AN x>0, y3,0,2%, JJS 1-8242 dzdady Volume Z-0 =)) (1-2242) lady changing in Palur form x = reaso, y=&sino, dady = rdodo limit of 8 is otot 27 aos y so P limit of ois oto I Then 0 2 O projection in ny plane )) (1.82gr, dady jsou ve vlody ) (8-8) do ( - ) - 7 1 - - 2 ) = 1 x 2-) 4 - T Answer 8 Canned with CamScanner

12 2 = 1-224² E 2=0 y 1-42 42 (x2+42) dz dady JUJ (x2+42 / dadydz aro E 1-2²-42 Indy =)) 1 21²+7?) [2] ) (x2442) (1-2²42) dady plane changing in Polar form: Projection of Ein ny- u= reaso, y = 8 sino, dady=&dodo 7x 072-0 limit of r is oto 1 8 limit of ois o to 25 Then U"/ 82(1-828 do do J) 1874 2) (1 - 2² 42) dady O 0 cscati CamScanner

Let x+y=1 u limit of u is oto 3 7 x-y = v limit of vis oto 2 2x = Utv с к Utv 2 u- y utv 2 y u-v 2 321 Ju IN JV dudes horup Jy he AC Ju ܝܐܝܢ dudr nt st - ایم (1 t + 4 + 1) dudy dady - chaty CS

Then ay2 dA I laty) e Ray alty) (ary) I latyle dady Ray 3 2. eurdodu - u e 2 N=0 u=0 2 3 uv du 4 e t Vzo 0 3 du (624-1) To 24 e 2 st 2 - 3 - 2 - 1 (1 - 3-2) 7 - e 4 4 = 7- ) Ansur Ics ante