(-1,0)区间上,根号下1减[x的平方] 的定积分 等于什么?
问题描述:
(-1,0)区间上,根号下1减[x的平方] 的定积分 等于什么?
答
令x = sinθ,dx = cosθdθ
x∈[-1~0] → θ∈[-π/2~0]
∫(-1~0) √(1 - x²) dx
= ∫(-π/2~0) cos²θ dθ
= (1/2)∫(-π/2~0) (1 + cos2θ) dθ
= (1/2)[θ + 1/2 • sin2θ] |(-π/2~0)
= (1/2)[0 - (-π/2 + 0)]
= π/4