Question

如何用定积分求解椭圆的面积以及球的体积

请给出推导的全过程 谢谢

Answer 1

S/4=ab∫(π/2，0）sint（cost)'dt=ab∫（0，π/2）sin�0�5tdt=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�5/0]（R^2-x^2-y^2)�0�5 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