Chapter 12, Problem 1PS

Cornu Spiral The cornu spiral is given by

$\begin{array}{ccc}x(t){\displaystyle {\int }_0^t\cos \left(\frac{\pi u^2}{2}\right)du}& \text{and}& y(t){\displaystyle {\int }_0^t\sin \left(\frac{\pi u^2}{2}\right)du}.\end{array}$

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 $t=0$ 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?