Chapter 12, Problem 1PS

Cornu Spiral The cornu spiral is given by

$x(t)={\displaystyle {\int }_0^t\cos \left(\frac{\pi u^2}{2}\right)du\text{and}y(t)={\displaystyle {\int }_0^t\sin \left(\frac{\pi u^2}{2}\right)du}}$

The spiral shown in the figure was plotted over the interval $-\pi \le t\le \pi$.

(a) Find the arc length of this curve from $t=0\text{to}t=a$.

(b) Find the curvature of the graph when $t=a$.

(c) The cornu spiral was discovered by James Bernoulli. He found that the spiral has an amazing relationship between curvature and arc length. What is this relationship?