Question

Suppose you have to use spherical coordinates to evaluate the triple integral SI z dV where D is the solid region that lies inside the cone z = 22 + y2 and inside the sphere 22 + y2 +22 = 144 D Then the triple ingral in terms of spherical coordinates is given by Select all that apply p3 cos • sin o dp do do D [!] > av = 6*6** ? [!] > av = 6"* )*S" So*%*%** 5% cos osin dp do do fp = av = 6**." [!] = 6* ** %* 12 p3 cos o sin o dp do do D z dV = تی و cos o sin o dp do do D

Answer 1

SSS z dv D cone z = 2 2 xty + 2 = 144. wheroe D is the solid segion that lies inside the J2²ry? and inside the spherse and the sphere 22+ y2+22 - 144 interesect when 22² + y 2) ²x+y = 72 22 The cone Z = Naty 2 2 = 144 Then 572 172-2² √144 - 2² y² SSS20V = zazdyda x=-572 y=-572-2² 2 = (x² + y2 s S D

Let x= e sino coso y = e sinq sino, 2= e cost Then e² 2² + y² + 2² and 312,4,2) ӘУ әe 22 ге oleo, o) 2x al 2x 20 da 20 22 ay 20 ay 20 ЭФ 22 20 sina coso sing sino 2009 ecoso caso e coso sino -esing esino coso lesina sine = p² sino coso sino coto sino - tang sino 2010 - esimo cose cool 60320 + sin o) + ton 0 (003?o + 3in2013 =p²in2a cosa [cota + tang] =p²sin? up coap x + sin?g sing coap = e? sino coa²p+ Х

ii d2dydx = 1 Mod dedo de 2(2,9, 2) 21 ,0,0) =p²sing dedo de Now x + y + z = 144 > p²=144 P=12 [:e7o i ose<12 z= √ 2 ² + y2 » ecoso = √ p²in2 cos²e +e? sino sin? lesina > ecoso - P sing 11 > tang = 1 4. Z=12 and P = 12 gives 12= 12 cosa cosp=1 7 9 = 0 .. 0< p < IT and 0 0 < 2 TT 4

Therefore SSS z dv 21. A 12 S s ecosp. e²sing dedo do 0 0 9 0 P=0 e cosa sing dedo do D 0