Question

∫(0到1) 根号(1-x^2)dx 怎么算 帮忙算下咯 谢谢了 在线等了 辛苦辛苦了

Answer 1

令x = sinθ，dx = cosθdθ
∫(0~1) √(1 - x²) dx
= ∫(0~π/2) cos²θ dθ
= (1/2)∫(0~π/2) (1 + cos2θ) dθ
= (1/2)(θ + 1/2 • sin2θ) |(0~π/2)
= (1/2)(π/2)
= π/4

追问

为什么原来是 (0到1) 后面变成(0到π/2) 了

追答

x = sinθ
当x = 0，sinθ = 0 => θ = 0
当x = 1，sinθ = 1 => θ = π/2