∫ (x ^ 2 + y ^ 2 + Z ^ 2) ds, the integral surface is a closed surface enclosed by x ^ 2 + y ^ 2 + Z ^ 2 = a ^ 2 (x ≥ 0, y ≥ 0) and plane x = 0, y = 0

∫ (x ^ 2 + y ^ 2 + Z ^ 2) ds, the integral surface is a closed surface enclosed by x ^ 2 + y ^ 2 + Z ^ 2 = a ^ 2 (x ≥ 0, y ≥ 0) and plane x = 0, y = 0

As shown in the figure:
The whole closed surface can be divided into four parts
Σ = Σ1 + Σ2 + Σ3 + Σ4
∫∫∑ 1 (X & # 178; + Y & # 178; + Z & # 178;) ds, the surface is Z = 0
= ∫∫Σ1 (x² + y²) dS
= ∫∫D (x² + y²) dxdy
= ∫(0→π/2) ∫(0→a) r³ drdθ
= (1/8)πa⁴
∫∫∑ 2 & nbsp; (X & # 178; + Y & # 178; + Z & # 178;) ds, the surface is x = 0
= ∫∫Σ2 (y² + z²) dS
= ∫∫D (y² + z²) dydz
= ∫(0→π/2) ∫(0→a) r³ drdθ
= (1/8)πa⁴
∫∫∑ 3 & nbsp; (X & # 178; + Y & # 178; + Z & # 178;) ds, the surface is y = 0
= ∫∫Σ3 (x² + z²) dS
= ∫∫D (x² + z²) dzdx
= ∫(0→π/2) ∫(0→a) r³ drdθ
= (1/8)πa⁴
∫∫∑ 4 & nbsp; (X & # 178; + Y & # 178; + Z & # 178;) ds, z = √ (A & # 178; - X & # 178; - Y & # 178;)
= ∫∫Σ4 a² dS
= a² * (1/8)(4πa²)
= (1/2)πa⁴
∴∫∫Σ (x² + y² + z²) dS
= 3 * (1/8)πa⁴ + (1/2)πa⁴
= (7/8)πa⁴