點樣可以integral一個surface既area?

S = {(x,y)|x²+y²=4²} : ∫∫ (xy) ds over Region S,

Let f(x, y, z) = xy, g(x, y) = sqrt(1 - x² - y²)

∂g/∂x = -x / sqrt(1 - x² - y²)
∂g/∂y = -y / sqrt(1 - x² - y²)

1 + (∂g/∂x)² + (∂g/∂y)² = 1 / (1 - x² - y²)

∫∫ f(x, y, z) ds over Region S, R : 1/4 unit circle
=8 ∫∫ f(x, y, g(x, y)) sqrt[1 + (∂g/∂x)² + (∂g/∂y)²] dA over Region R
=8 ∫∫ (xy) sqrt[1 / (1 - x² - y²)] dA over Region R
=8 ∫∫ (r cos θ)(r sin θ) sqrt[1 / (1 - r²)] r dr dθ over Region { (r, θ) : r ∊ (0, 4), θ ∊ (0, π/2) }
=8 ∫∫ (r³ sin (2θ) sqrt[1 / (1 - r²)] dr dθ over Region { (r, θ) : r ∊ (0, 4), θ ∊ (0, π/2) }
=8 ∫ (r³ sqrt[1 / (1 - r²)] dr ∫ sin (2θ) dθ over Region { (r, θ) : r ∊ (0, 4), θ ∊ (0, π/2) }

2007-05-09 00:04:17 補充：
我好似用錯了 formula 計，未刪除就已經.............