计算曲面积分如图其中曲面是柱面x^2+y^2=1被平面z=0和z=3所截得的在x》=0的部分,取外侧

高斯公式法.
取Σ：x² + y² = 1,前侧
补Σ1：z = 3,上侧
补Σ2：z = 0,下侧
补Σ3：x = 0,后侧
∫∫(Σ+Σ1+Σ2+Σ3) ydzdx = ∫∫∫Ω (0 + 1 + 0) dxdydz
= ∫∫Ω dxdydz
= (1/2) * π * 1² * 3
= 3π/2
∫∫Σ1 ydzdx = ∫∫Σ2 ydzdx = ∫∫Σ3 ydzdx = 0
所以∫∫Σ ydzdx = 3π/2
普通法.
Σ：x² + y² = 1,前侧
取Σ1：y = - √(1 - x²),左侧
取Σ2：y = √(1 - x²),右侧
∫∫Σ ydzdx
= ∫∫Σ1 ydzdx + ∫∫Σ2 ydzdx
= - ∫∫D [- √(1 - x²)] dzdx + ∫∫D [√(1 - x²)] dzdx
= 2∫∫D √(1 - x²) dzdx
= 2∫(0,3) dz ∫(0→1) √(1 - x²) dx
= 2 * (3 - 0) * (1/4)(π)(1⁵)
= 3π/2