Question

高等数学～曲线积分～

Answer 1

答：3πa²

参数法：
x² + y² + z² = a²
x = y
设 x = y = a cost
设 z = a sint
ds = √(x'² + y'² + z'²) dt
= √(a²cos²t + a²cos²t + a²sin²t) dt
= a√(1 + cos²t) dt
所以
∮_(Γ) √(2y² + z²) ds
= ∫(0,2π) √(2a²cos²t + a²sin²t) * a√(1 + cos²t) dt
= a²∫(0,2π) √(1 + cos²t) * √(1 + cos²t) dt
= a²∫(0,2π) (1 + cos²t) dt
= a²(2π) + a²∫(0,2π) (1 + cos2t)/2 dt
= 2πa² + (a²/2)(t + (sin2t)/2) |(0,2π)
= 2πa² + (a²/2)(2π + 0)
= 2πa² + πa²
= 3πa²