How do I find the length of the polar spiral $r = a\cdot \theta^2 - 1 + b\cdot \theta$?

Question

How do I find the length of this polar spiral?

$$r = a\cdot \theta^2 - 1 + b\cdot \theta,$$ where $a$ and $b$ are variables.

Answer 1

The formula tells us ( assuming $\;0\le\theta\le2\pi\;$ )

$$\mathcal L=\int_0^{2\pi}\sqrt{r^2+r_\theta^2} \,d\theta=\int_0^{2\pi}\sqrt{(a\theta^2+b\theta-1)^2+(2a\theta+b)^2}\,d\theta$$

Not the nicest integral to see...but give it a try.

Pay attention to the fact that

$$r_\theta^2=\left(\frac{dr}{d\theta}\right)^2$$