Question

3. Consider the triple integral , g(x, y, z)dV, where E is the solid bounded above by the sphere x2 + y2 + z2 = 18 and below by the cone z= x2 + y2. a) Set up the triple integral in rectangular coordinates (x,y,z). b) Set up the triple integral in cylindrical coordinates (r,0,z). c) Set up the triple integral in spherical coordinates (0,0,0).

Please explain steps

Answer 1

3) Here the tripple integral is given as, solid bounded =S (2522) dv phere x² + y² + z ²2 18 and La Where E in the E abore by the 21) teloo by the cone z² = x²+y? てん) we can consider thre cone AZ z?ny*+7's 16 Ez hezy2 Zz 5x²ty? need consider the pout pout above ny plane only [~ Sphere in the above above sanface and below surface mentioned] (*) Now, in calitesiam Co-ordinates ra²ty? from a to ribor-ye (up to sphere Core is Z races 18 \ مواليد Also, (1) and ce intersect at z22² =) =9, Z=3 [ Zo) So, the domain of ay x²+y²=9 Cas shown in figure] from g-x².to So, Verry from from ²-3 to 3 18-x=y2 I: geasy,z) axayaz y should Varry and xe will 3 x² 7x SIS Xus-3 yo (6) now, X ) Zz Patyz EFN) x2 [vd V2 dxdy dz Now, ton cylindrical co-ordinates (6,0,2) we know, p²2 xzy} oz tomia e drz pdo doda Now, & varies from to [y z²zny? p? to N18-82 Vio o varies from o to 29 need to tum 360°, finally from the projection xy plane we fina no varies from o to 3 whole to

So, Iz SSS peo e zo zer as a²+y? =3 has form 123 m sus plane 18-0- G. (rosa) o dedoar [where, gex, y, z) z G, (8,0,7)] (C) Finally in spherical polar co-ordinates (184) p² x² + y² + z² tanoz x?ty? ånd xap sino Candy zo sino sind, zzoplas Horse raneoad and Hero p varies from a to N18 [ the surface enclosing in there of radius N78] : & mest varry from s to za Now, the semiventical angle of Cone must to o; so, from the information za standz x 223 3 Zz3 en a pan8 ve get 2 5T8 El so must 2T nis Caso ₂ rany from o to Tyy Hence, I? G2 (1,0,0) esmo de dado 0 2002 [ Also [ dv? pa smo do do do here] g(x4,2) 22,00) This completes the answer. S=0