计算函数y=130cos(x*pi/1000)一个周期的曲线长度?

Question

计算函数y=130cos(x*pi/1000)一个周期的曲线长度?

Answer 1

ds=sqrt(dx^2+dy^2)=sqrt(1+y'^2)dx=sqrt(1+(130Pi/1000sin(x Pi/1000))^2)dx
积分
s=integrate[sqrt(1+(130Pi/1000sin(x Pi/1000))^2),{x,0,2000}]
=Pi/1000*integrate[sqrt(1+(0.130Pi)^2sin(y)^2),{y,0,2Pi}]
这个积分需要用到复变函数z=e^（iy）对x积分一周期就是对z积分一周
s=Pi/1000*integrate[sqrt(1+(0.130Pi)^2 ((z-z^(-1))/2)^2)/iz,{z,circle}]
=Pi/1000*integrate[sqrt(z^2+w)/iz^2,{z,circle}]
w=(0.130Pi)^2 ((z^2-1)/2)^2