Question

hi this is a calc 3 problem. i dont know where i went wronf when i tried to solve it. please help and thank youuu

Use the divergence theorem to find the outward flux of F across the boundary of the region D. F= -x?i+3x?yj + 6xzk D: The region cut from the first octant by the sphere x² + y2 + z = 4 91 The outward flux is 2 (Type an exact answer, using a as needed.)

Answer 1

cut from the first octant by S V Given that F = -43 h+3x?yj +622k D: The region the sphere x²xy + 2 = 4 By Gauts divergence the dom SS Foños-5156.pdv The solid D bounded by x²ty tz² = 4 in the first octant. In the spherical polar cool dinates, dV= 12 sin go dire do do osrsa , 02.0572 ospel2 : Dis only in the first Sai + Sy stok) • (-78e +3Ⓡyj +622k) 2 and octant OF

a (-43) + 2 (3x²y) + az dx & (622) 7. F and product of (By the definition 3* + 362 aanx +62 13 ox Spherical F the limits of 2 (ay)= x (nynt) dy and & (x2") = x (M2M) V.F = 6x Substituting the coordinates ela e Ceso simp, Y = 92 Smolim, Z= R. Caso 6 (r.coso smp) Substituling VF and using r, o and P in the first octant, wegel- SSF.nds - SSS V.F dv 자 지 오 C6 mesos O Simp) 12 dredo de (Simo? 0-0 2 2 2 S 56 93 caso sono dite do do = SAT2 cos co sm?go do do : Stryn 2 0-0 NEO - 0-0 Bro 900 6 ! poo Oro 72 case Sm?y do dp Do OCO

2 6 (2²) Įsmo] 2 sin odpr: Jesodo sime O peo X2 2 0-0 b 24 ( Sim T - Smo) (1-Cos29) dp SfCo do F(b) – F(a) Sim² = 1- Cosap a and 2 - 24 I (1-ceszq) do Sm A = 7 Smo = 0 0=0 7/2 - 2 ſp- sinze 0 -ot smo) . 12 ( I - 3m 2. (V) 2 Sia)- Sm(0) = 0 c 12(♡ GT 6T SS F.ñ ds = 555 0.F dv - S V