求心形线{π,2π}的长度,为什么结果为负数?心形线为:p=a*(1+cosθ)

Question

求心形线{π,2π}的长度,为什么结果为负数?
心形线为:p=a*(1+cosθ)

Answer 1

L=∫(π,2π)sqrt[ρ^2+(ρ')^2]dθ
=∫(π,2π)sqrt[a^2(1+cosθ)^2+a^2(sinθ)^2]dθ
=a∫(π,2π)sqrt(2+2cosθ)dθ
=2a∫(π,2π)|cos(θ/2)|dθ
=2a∫(π,2π)[-cos(θ/2)]dθ (*)
=4a[-sin(θ/2)]|(θ=2π)-4a[-sin(θ/2)]|(θ=π)
=0-(-4a)
=4a
(sqrt=根号)
如果算得负数应该是去绝对值时忘了加负号,(*)那一行.