不用定积分如何求椭圆面积

Question

如题所述

Answer 1

S/4=ab∫(π/2,0）sint（cost)'dt=ab∫（0,π/2）sin?tdt=π/4*abS=πab 令 y = √(R^2-x^2) sint ,dy = √(R^2-x^2) cost dt,t :[ 0,π/2 ]
∫[0,R] dx ∫ [(R^2-x^2)?/0]（R^2-x^2-y^2)?dy
= ∫[0,R] dx ∫ [ 0,π/2] ( R^2 - x^2) (cost)^2 dt
= ∫[0,R] ( R^2 - x^2) dx ∫ [ 0,π/2 ] (1+cos2t)/2 dt
= ( π/4) ∫[0,R] ( R^2 - x^2) dx
= ( π/4) * 2R^3 /3 = πR^3 /6
V = 8 * πR^3 /6 = 4πR^3 /3

Answer 2

Answer 3

用公式吧πab