Question

求曲面积分∫∫zds期中∑为抛物面z=2-(x^2+y^2)在xoy面上方的部分

答案是37π/10

Answer 1

Σ：z = 2 - (x² + y²) ==> x² + y² = 2 - z、开口向下。上侧
dz/dx = - 2x、dz/dy = - 2y
∫∫Σ z dS
= ∫∫D [ 2 - (x² + y²) ] √(1 + 4x² + 4y²) dxdy
= ∫(0→2π) dθ ∫(0→√2) ( 2 - r² )√(1 + 4r²) * r dr
= (2π)(37/20)
= 37π/10