3.6 Rectifiable curves

Using the methods of this course, one can work out the integral for $L$ and find an explicit formula for arclength $s(t)$ along the following curves:

$${\hbox{circle}}\qquad x^{2}+y^{2}=a^{2},$$

$${\hbox{parabola}}\qquad y^{2}=4ax,$$

$${\hbox{Neil's curve}}\qquad y^{2}=x^{3},$$

$${\hbox{Tsirnhausen's cubic}}\qquad 3y^{2}=x^{2}(1-x),$$

$${\hbox{catenary}}\qquad y=\cosh x,$$

$${\hbox{logarithmic spirals}}\qquad r=ae^{b\theta}\;{\hbox{(in polar coordinates)}},$$

$${\hbox{astroids}}\qquad|x|^{2/3}+|y|^{2/3}=1.$$