数学定积分应用，计算面积？

Question

谢谢~

Answer 1

y=(1/2)x^2
x^2+y^2 =8

x^2 + (1/4)x^4 =8
x^4+4x^2 -32=0
(x^2+8)(x^2-4)=0
x=2 or -2
let
x=2√2 sinu
dx =2√2 cosu du
x=0, u=0
x=2 , u=π/4
A
= ∫(-2->2) [ √(8-x^2) - (1/2)x^2 ] dx
= ∫(-2->2) √(8-x^2) dx - (1/6)[x^3 ] |(-2->2)
= ∫(-2->2) √(8-x^2) dx - 8/3
=2∫(0->2) √(8-x^2) dx - 8/3
=16∫(0->π/4) (cosu)^2 du - 8/3
=8∫(0->π/4) (1+cos2u) du - 8/3
=8[u+(1/2)sin2u]|(0->π/4) -8/3
=4(π/4 + 1/2) -8/3
=2π + 2 -8/3
=2π -2/3
另外一个的面积
=圆面积 - A
=8π -[2π -2/3]
=6π +2/3