Question

10. (15 points) Evaluate the integral SSSE FDV with f = y2-2. E is the solid in the first octant that lies above the cone 2 = V x2 + y2 and below the sphere x2 + y2 + 22 = 1.

Answer 1

solution. We have to evaluate integral for with f = 9²22 E: is solid in first octant that lies above the cone z= √x²+42 and below the sphere x²+4² + 2 = 1 to Now we use the spherical co-ordinate evaluate the integral Now, Replace x = rood sino Y = rsino. sind z = roop where limits of ro, and & are. OLILI, OCOLI and Topo=fr 2 sin (sin otupo) & scopo= rsing → 48 = 1 + 0 = I so ocoLT Now dr = r2 sino drdo.do cru2, 2 dv Hance (Star = 11( 4 222 dv Hence fav= T T1121 S o OOO sino sin p. 10²0 dr do do

T14 12 Ifid sind. wxo do do 14 Til2 sinho sin34. copo do.de Tl14 T1/2 (4-10p20) sinbp cop?p.48.00 7714 Lal 2 (449-491722°.com?4 do 1/4 sin3&. cop?4 do - I let cope=t - sind do= dt 161-42)£dt when t=0&t=1 8-11 > t = 1/2 vasts] + = Wit 78 3163441487 05(*)]

f Hence! If dr = Il 3 - I 28 1 is 6052 where f= y 2₂2 ans