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.

Do you know a formula for arc length using polar coordinates? — Zubin Mukerjee, Jan 21 '19 at 12:07
https://www.khanacademy.org/math/integral-calculus/ic-adv-funcs/dc-polar-arc-length/v/polar-arc-length-formula — For the love of maths, Jan 21 '19 at 12:11

score 1 · Answer 1 · answered Jan 21 '19 at 12:11

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

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

answered Jan 21 '19 at 12:11

DonAntonio

I've no idea. I haven't even tried to solve that integral: it looks nasty. Yet you can try to do it with Wolfram or some other program... – DonAntonio Jan 23 '19 at 12:50
@user33915 I'm sorry but I still have no idea, yet this time it is because I can't understand what you mean: why the integral changes and all?? And how come you want to integrate with the variable ($,\theta,$) being fixed? – DonAntonio Jan 23 '19 at 13:53

1 Answers1