Parametrizing a counterclockwise spiral

Question

I am looking for someone who could help explaining the solution to this question? Especially how they came up with the paramaterization and why it would be cos(t+ pi/2) and sin(t + pi/2) instead of cos(t) and sin(t).

Thanks a lot!!

Answer 1

Your question about the phase shift of $\pi/2$ is very easy to understand if we look at this problem in the complex plane. Let

$$z(t)=t^2e^{i(t^2+\pi/2)}=t^2e^{\pi/2}e^{it^2}=it^2e^{it^2}$$

So, the difference between $t^2$ and $(t^2+\pi/2)$ boils down to a factor of $i$, which translates to a $90^{\circ}$ CCW rotation of $z$. In fact, any phase shift will translate to a rotation of that amount in the complex plane. In your problem, if the $\pi/2$ is omitted, then the $4\pi$ point would fall on the $x$-axis rather than the $y$-axis. Continuing along with the complex plane, the arc length is given by

$$s=\int |\dot z|dt$$

$$\dot z=i[it^2+2t]e^{it}=(-t^2+i2t)e^{it}$$

$$|\dot z|=\sqrt{t^4+4t^2}=t\sqrt{t^2+4}$$

and finally, as shown by Wilson

$$s=\int_0^{2\pi} |\dot z|dt=\frac{(t^2+4)^{3/2}}{3}\big{|}_0^{2\pi}=\frac{8}{3} \left((1+\pi^2)^{3/2}-1\right)$$

I find that many problems are much more easily formulated, solved, and understood in the complex plane.