求定积分∫ 根号(1-x^2)dx(上下限0—1⼀2)

2024-11-06 02:50:03
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回答(1):

设x = siny,dx = cosy dy
当x = 0,y = 0;当x = 1/2,y = π/6
∫(0→1/2) √(1 - x²) dx
= ∫(0→π/6) √(1 - sin²y) • cosy dy
= ∫(0→π/6) cos²y dy
= (1/2)∫(0→π/6) (1 + cos2y) dy
= (1/2)(y + 1/2 • sin2y) |(0→π/6)
= (1/2)(π/6 + √3/4)
= (3√3 + 2π)/24