这是第一类曲面积分。
z = ±√(a^2-x^2-y^2),
I = ∯z^2dS = 2 ∫∫ z^2√[1+(z)^2+(z)^2] dxdy
= 2 ∫∫ (a^2-x^2-y^2)a/√(a^2-x^2-y^2) dxdy
= 2a ∫∫ √(a^2-x^2-y^2) dxdy
= 2a ∫dt ∫ √(a^2-r^2) rdr
= -2πa ∫ √(a^2-r^2) rd(a^2-r^2)
= -2πa [(2/3)(a^2-r^2)^(3/2)] = (4/3)πa^4
这是第一类曲面积分。
z = ±√(a^2-x^2-y^2),
I = ∯z^2dS = 2 ∫∫ z^2√[1+(z)^2+(z)^2] dxdy
= 2 ∫∫ (a^2-x^2-y^2)a/√(a^2-x^2-y^2) dxdy
= 2a ∫∫ √(a^2-x^2-y^2) dxdy
= 2a ∫dt ∫ √(a^2-r^2) rdr
= -2πa ∫ √(a^2-r^2) rd(a^2-r^2)
= -2πa [(2/3)(a^2-r^2)^(3/2)] = (4/3)πa^4