Question

微积分 求解

Answer 1

这是过程

Answer 2

解：在球面坐标系中计算：
ʃʃʃv(x/a+y/b+z/c)²dV
=ʃ[0,2π]dθʃ[0,π]dφʃ[0,R](ρsinφcosθ/a
+ρsinφsinθ/b+ρcosφ/c)²ρ²sinφdρ
=[(R^5)/5]ʃ[0,2π]dθʃ[0,π](sinφcosθ/a
+sinφsinθ/b+cosφ/c)²sinφdφ
=[(R^5)/5]ʃ[0,2π]dθʃ[0,π][(cosθ/a
+sinθ/b)sinφ+(1/c)cosφ]²sinφdφ
=[(R^5)/5]ʃ[0,2π]dθʃ[0,π][(cosθ/a+sinθ/b)²sin³φ+(2/c)(cosθ/a+sinθ/b)sin²φcosφ
+(1/c²)cos²φsinφ]dφ
=[(R^5)/5]ʃ[0,2π][(cosθ/a+sinθ/b)²(4/3)
+0+(1/c²)(2/3)]dθ
=[4(R^5)/15]ʃ[0,2π][(cosθ/a+sinθ/b)²
+1/(2c²)]dθ
=[4(R^5)/15]ʃ[0,2π][cos²θ/a²+sin2θ/(ab)+sin²θ/b²+1/(2c²)]dθ
=[4(R^5)/15][π/a²+0+π/b²+2π/(2c²)]
=[4π(R^5)/15](1/a²+1/b²+1/c²).