是锥面z=√x^2+y^2的下侧,计算曲面积分∫∫xdydz+2ydzdx+3zdxdy

Question

是锥面z=√x^2+y^2的下侧,计算曲面积分∫∫xdydz+2ydzdx+3zdxdy

Answer 1

还缺条件,假设Σ是z = √(x² + y²)和z = h (h > 0)围成.补面z = h(上侧)
∫∫(Σ+Σ1) xdydz + 2ydzdx + 3zdxdy
= ∫∫∫Ω (1 + 2 + 3) dV
= 6∫∫∫Ω dV
= 6∫(0,2π) dθ ∫(0,h) rdr ∫(r,h) dz
= 6 * 2π * ∫(0,h) r * (h - r) dr
= 12π∫(0,h) (hr - r²) dr
= 12π(h * h²/2 - h²/3)
= 2πh³
∫∫Σ1 xdydz + 2ydzdx + 3zdxdy
= 3h∫∫Σ1 dxdy
= 3h∫∫D dxdy
= 3h∫(0,2π) dθ ∫(0,h) rdr
= 3h * 2π * h²/2
= 3πh³
于是∫∫Σ xdydz + 2ydzdx + 3zdxdy + 3πh³ = 2πh³
==> ∫∫Σ xdydz + 2ydzdx + 3zdxdy = - πh³
其中h那个你还未提供.