设心脏线方程为r=1+cosθ,求心脏线围成图形面积,求心脏线的长度

【参考答案】
r＝1+cosθ,r'＝-sinθ
利用对称性
长度=2∫(0,π)√r^2+r'^2dθ
=2∫(0,π)√(2+2cosθ)dθ
=2∫(0,π)√4cos^2(θ/2)dθ
=4∫(0,π)cos(θ/2)dθ
=8∫(0,π)cos(θ/2)dθ/2
=8sin(θ/2)|(0,π)
=8
面积=2*1/2∫(0,π)r^2dθ
=∫(0,π)(1+cosθ)^2dθ
=4∫(0,π)cos^4(θ/2)dθ
=8∫(0,π)cos^4(θ/2)dθ/2 (令θ/2=t)
=8∫(0,π/2)cos^4tdt
=8*3/4*1/2*π/2
=3/2*π