Chapter 10, Problem 7PS

(a)

To determine

T graph: The provided parametric functions $x\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}\cos \frac{\pi u^2}{2}du}$ and $y\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}\sin \frac{\pi u^2}{2}du}$ over the interval $-\pi \le t\le \pi$.

(b)

To determine

To prove: The Cornu spiral formed by the function $x\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}\cos \frac{\pi u^2}{2}du}$ and $y\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}\sin \frac{\pi u^2}{2}du}$ where $-\pi \le t\le \pi$ is symmetric with respect to origin.

(c)

To determine

To calculate: The length of the Cornu spiral from $t=0$ to $t=a$. Also, find the length of the spiral from $t=-\pi$ to $t=\pi$.