Evaluate the surface integral x2 y2 z2 dS S is the part of

Evaluate the surface integral.

x² + y² + z²

dS
S is the part of the cylinder

x² + y² = 9

that lies between the planes

z = 0 and z = 4,

together with its top and bottom disks

S

Solution

We do this in three parts.

1) The cylinder x^2 + y^2 = 9 with z in [0, 4].

Parameterize via R(u, v) = <3 cos u, 3 sin u, v> for u in [0, 2], v in [0, 4].

Since R_u x R_v = <-3 sin u, -3 cos u, 0>,

we have ||R_u x R_v|| = 3.

Hence, (x^2 + y^2 + z^2) dS

= (v = 0 to 4) (u = 0 to 2) (9 cos^2(u) + 9 sin^2(u) + v^2) * 3 du dv

= (v = 0 to 4) 2 (27 + 3v^2) dv

= 2 (27v + v^3) {for v = 0 to 4}

= 344.

2) The bottom disk x^2 + y^2 9 with z = 0.

Parameterize via R(u, v) = <v cos u, v sin u, 0> for u in [0, 2], v in [0, 3].

Since R_u x R_v = <0, 0, -v>, we have ||R_u x R_v|| = v.

Hence, (x^2 + y^2 + z^2) dS

= (v = 0 to 3) (u = 0 to 2) (v^2 cos^2(u) + v^2 sin^2(u) + 0^2) * v du dv

= (v = 0 to 3) 2v^3 dv

= v^4/2 {for v = 0 to 3}

= 81/2.

3) The top disk x^2 + y^2 9 with z = 4.

Parameterize via R(u, v) = <v cos u, v sin u, 0> for u in [0, 2], v in [0, 3].

Since R_u x R_v = <0, 0, -v>, we have ||R_u x R_v|| = v.

Hence, (x^2 + y^2 + z^2) dS

= (v = 0 to 3) (u = 0 to 2) (v^2 cos^2(u) + v^2 sin^2(u) + 4^2) * v du dv

= (v = 0 to 3) 2 (v^3 + 16v) dv

= (v^4/2 + 16v^2) {for v = 0 to 3}

= (81/2 + 144).

Thus, the net surface integral equals

344 + 81/2 + (81/2 + 144) = 569.