Surface Integrals

The surface y =3√(x^2+z^2) , y=3~9 can be parametrized as
R=(x,y,z)=(rcosθ, 3r, rsinθ), r=1~3, θ=0~2π
dS=| ∂R/∂r x ∂R/∂θ| dr dθ = √10 r dr dθ
∫∫_S y dS
=∫[0~2π] ∫[1~3] 3r*√10 r dr dθ
=52π√10

2011-07-21 22:49:22 補充：
(b)integrate directly.
R=(x,y,z)=(rcost, rsint, 3-rcost-rsint), r=0~6, t=0~2π
dS= ∂R/dr x ∂R/dt dr dt= (r, r, r) dr dt
∫∫_s F dot dS
=∫[0~2π] ∫[0~6] [3r^2 cost+ r - 2(3-rcost-rsint) r ] dr dt
=∫[0~2π] [216 cost- 90+ 144(sint+cost) ] dt
= -180π

2011-07-21 23:41:23 補充：
ydS=y* dx dz/|cosβ| = y dx dz/(2y) *| grad(9x^2-y^2+9z^2) |
= √(81x^2+y^2+81z^2) dx dz=3√10 *√(x^2+z^2) dx dz
∫∫s ydS=∫∫s_zx 3√10 *√(x^2+z^2) dx dz
=∫[0~2π]∫[1~3] 3√10 *r * rdr dt
=52√10*π

2011-07-21 23:59:32 補充：
Projection 的方法沒有比較好