Question

Making a change to spherical coordinates, evaluate the integral

Haciendo un cambio a coordenadas esféricas, evalúe la integral 14- S. 4–32-y? 1 dzdady 224 72122 OT/4 OT OT2 O NO ESTÁ LA RESPUESTA

Answer 1

2 Given √4-y² 4-x²y² y 21 y 7 y z dzdxdy 0 Using spherical Coordinates X= 8 sing loso; y=fsind sino ;z=flos & and x² + y² + z ² = 82 Limits of integration are Z - dimits oeze JA-x²yz

from upper Z-limit Z=4-x 2 y2 z2=4-x?y? + x2+42 + z²4 x² + y² + z ² = 82=4 p= A=2 X-limits it oLX 4-y? from upper X- limity x=√4-y2 * x² + y² = 4 y- limits → o Ly (2 rondisson Grouphing the x - and y limits ay bounded region sx=(4-y? #x²4y²4 >x 2 N. -2 from the above graph o- limits are O limits : ہے 8 7 6 T प

from lowed 7- limit Z=0 glosø=o - Cost=0 Р → Ø = Cost(0)=17 =>ø= 1 - Ø-limits a= Ø E JA-ya St-x²y² 2 u x 4 y ²1 22 dzdx dy 1/2 T/2 2. [ Paz (pring) padeda Lee [ x?ty 2+ z² = pz 77/2 TT/2 2 sino de dado TV/2 TV 2 [s] dede $ 5 sing [s] to y sino [2-o] dede -

T/2 TY2 = 2 Sings [o]," do TV/2 11 S sing [11 - oj do T [- coss] 2 $ -TO [ cos( 7 ) cos(o) =-71 (0-1) - TT 2 Ay2 SA-x²y2 So s dz x²ty²+72 dady Answer = π