简单方法推导椭圆面积公式我微积分很差劲啊,还有三角函数也有点晕,有没有简单办法推导出 πab是咋来的?

设椭圆方程为x=acost,y=bsint,则椭圆的面积S=[0,π/2]4∫ydx=[0,π/2]4∫bsint(asint)dt
=[0,π/2]4ab∫sin²tdt=[0,π/2]4ab∫[(1/2)(1-cos2t)]dt=[0,π/2]2ab[∫dt-(1/2)∫cos(2t)d(2t)]
=2ab[t-(1/2)sin2t]∣[0,π/2]=2ab(π/2)=πab