Question

8. Write a triple integral including limits of integration that gives the volume between the top portion of the sphere x2 + y2 + 2 = 9 and the plane z = 2. Evaluate the integral. 9. Calculate the line integral fĒ. dr where F(x, y) = (x + y) 7 + (x+y); and C is the path given by r(t) = (t) 7 + (t?); for 0 <t<l.

Answer 1

so Z (0,0,3) sayt z 3² 2-2 ty y The given region is bounded abone by x + y +2=9 and bounded below by z=2 the find the triple integral of volume by spherical method. where, nel caso sino se sino sind zsecond where e in the position vector, o is the angle between projection of e on my plane and a axis and & is the angle between z axis and e. x² + y² +229 e²=3 → e=3 Now, 088 do the bounds coso Now, 7=2 e cos q=2 = e = of e is, the bound çe < 3 and, In my plane, the range of o is whole ny plane, i e. of oL QT

2 Now, loneer bound of ¢ in o B 92 (A 3 and for upper bound, LA OB=P. & BO = 2,0A-3 .. cos * o acos cos (3) 22/35 ф2 сот 6) dva p² sind ode do do Then, volume Sslav Cosa 2 3 2T ost) e sind de de do i rose - 40" ("*** ET 10 AM sim (6 - 1w) cos 2Ã cost 3 cose ) ad do 9 sin 8 3 tand do do o 12T 25 cost 22 = jah [-9.45 log/sec * -9 608 (673)-{ logosco12) |+9.690 +0] do x1 ].com" 20 1 E 2a 8 log logle 1/2) + 9 do .27 do 3-8 Log(3) 5 - log(3)) 27 | duy