Question

面积分 如下第5题 同志们,速战速决,哥们,

Answer 1

S分为两部分,S1:y=√(R^2-x^2),S2:y=-√(R^2-x^2).

S1与S2在zox面上的投影区域都是D:-R≤x≤R,0≤z≤H.

dS=√[1+x^2/(R^2-x^2)+0]dzdx=R/√(R^2-x^2)dzdx

原积分

=∫∫(S1)dS/r^2+∫∫(S2)dS/r^2

=∫∫(D) 1/(R^2+z^2)×R/√(R^2-x^2)dzdx+∫∫(D)1/(R^2+z^2)×R/√(R^2-x^2)dzdx

=2∫∫(D) 1/(R^2+z^2)×R/√(R^2-x^2)dzdx

=2∫(0到H) 1/(R^2+z^2)dz ∫(-R到R) R/√(R^2-x^2)dx

=2×1/R×arctan(H/R)×R×π

=2πHarctan(H/R).