Question

HW 4(II) - Triple Integrals (Cylindrical+Spherical) (1) Sketch E and then use cylindrical coordinates to evaluate /// f(x,y,z) dv. (@) 12,90 (b) y (c) f(x, y, z) = y; E:x2 + y2 <1,x > 0, y = 0,05252 (d) f(x,y,z) = x E: x2 + y2 <z 59

Answer 1

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Ans (0) fex, y, z)=y do +4321, 220, 220, 04222. 42 110,0,2) 20,1,0) -29 IS try, a (1,0,0) Now for spherical coordinates x=82030, y=8 Sino, 2=2 dx dy dz=&dodedz Hese limit of 2 will be

2-0 to 2-9 (as given os22 tos xy-limit, projection of the soled in ay plane is 22 +41 X20, 430 000 SX but X=81018, y=ssine in Xth1 =(8 lose) +(8 sino)? = 1 =) sz (cast 0 +8inte) <. a 841 ) 8 =) So limit of s will be 8=0 to 8=)

limit of 0 will be 0-0 to 0=0 so sss fasz) dx dydz 2-1, 2-2 L ) (8 sine) sdrdodz 0-08-02 2= 0 8=1 - S o sino [z] Ldr de 8 = 0 11 -26" El sinode 3S "sinode = 2[(050]) 3 50+13=3

o f12,3,2)=2, EX+53259 For spherical coordinates but X=52080, y=88ind, 2=2 dady_80sde de dy dz=&dade de So region will be x² + y2 <249 =) (2 Coso) + (8 sing <249. 8(05229 So limit of 2 will be 2-st to 2=9.

N 2:9 12-24h at 2-9, from a ty 5229 we get x2 + y2 <9. Prosi+=9 ) ? <g 853 So limit of 2 will be 8=0 to 8=3 limit of & will be 00 to 0-25

So fas, 2) dady dz 2118, 9 odo do dz O 02-82 27,3 0-0 o 12.

lol - OP 3P (35-19) * S ople op L32 lol L682 = 727 (1-4) (m) – 943 27