∫(0，2) x√(2x - x²) dx
= ∫(0，2) x√[- (x² - 2x + 1) + 1] dx
= ∫(0，2) x√[1 - (x - 1)²] dx
令x - 1 = sinθ，dx = cosθ dθ
x = 0 --> θ = - π/2
x = 2 --> θ = π/2
= ∫(- π/2，π/2) (1 + sinθ)|cosθ| * cosθ dθ
= ∫(- π/2，π/2) (1 + sinθ)cos²θ dθ
= ∫(- π/2，π/2) cos²θ dθ + ∫(- π/2，π/2) sinθcos²θ dθ
= 2∫(0，π/2) (1 + cos2θ)/2 dθ + ∫(- π/2，π/2) cos²θ d(- cosθ)
= [θ + (1/2)sin2θ] |(0，π/2) - (1/3)[cos³θ] |(- π/2，π/2)
= π/2
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