Question

椭圆面积用定积分算

x2/a2＋y2/b2＝1.所围成图形的面积，就是有一步是4∫（0，a）b/a√（a2-x2）dx中的b/a√（a2－x2）怎么来的，我怎么都算不出来？

Answer 1

x^2/a^2 + y^2/b^2 = 1
y = (b/a)√(a^2 - x^2)
面积 = 4∫(0→a) (b/a)√(a^2 - x^2) dx，令x = a*sinθ，dx = a*cosθ dθ
= (4b/a)∫(0→π/2) √(a^2 - a^2sin^2θ)*(a*cosθ dθ)
= (4b/a)∫(0→π/2) (a*cosθ)(a*cosθ) dθ
= (4ab)∫(0→π/2) cos^2θ dθ
= (2ab)∫(0→π/2) (1 + cos2θ) dθ
= (2ab)[θ + (1/2)sin2θ] |(0→π/2)
= (2ab)(π/2)
= πab

追问

椭圆用y用x表示怎么得的？我是b2x2＋a2y2=a2b2.y2=b2－b2x2/a2.开了根号以后和你的结果不一样呀

追答

怎会不一样？你是不会解这个根号吧？代入x = a*sinθ就好了