心形线周长

Question

心形线周长

Answer 1

r=a(1+cosθ)(a>0)
C=∫(r^2+r'^2)^(1/2)dθ 其中,r'表示r的导数,积分上限2π,下限为0
C=∫{[a(1+cosθ)]^2+(asinθ)^2}^(1/2)dθ
=a*∫[2+2cosθ)^(1/2)dθ
=2a*∫|cos(θ/2)|dθ=2a*[∫cos(θ/2)dθ (上限为π,下限为0)+∫-cos(θ/2)dθ(下限为π,上限为2π)]
=8a