Question

Find the exact length of the equiangular spiral.

Answer 1

\[r=e^\theta\]

Answer 2

theta from 0 to 2pi

Answer 3

formula \[\int\limits_{o}^{2\pi} (e^2\theta +e^\theta)^1/2 ~~~d \theta\]

Answer 4

I am not getting the correct answer

Answer 5

arc length ?

Answer 6

And from wolfram i am seeing hyperbolic function as part of the answer.

Answer 7

yes arc lenght = length

Answer 8

so in other to integrate we can factor out the \[e^\theta\]

Answer 9

\[\int\limits_{?}^{?}[e^\theta(e^\theta+1)]^1/2~~~~d \theta\]

Answer 10

And let u=e^theta +1

Answer 11

and i end up with \[2/3[(e^\theta+1)]^3/2\]

Answer 12

theta form 0 -2pi

Answer 13

what's the general formula for arc length in polar coords?
something like
\[L= \int \sqrt{ r^2 + \left(\frac{dr}{d\theta}\right)^2 } \ d\theta\]?

Answer 14

Yes.

Answer 15

oh so my formula is wrong then.

Answer 16

Am i suppose to integrate \[e^(4~\theta)\]

Answer 17

sqrt

Answer 18

and r^2= e^2x
dr/dx = e^x
and
(dr/dx)^2 = e^2x

so
\[ \int \sqrt{2e^{2x}} dx \\ \sqrt{2}\int e^x \ dx\]

Answer 19

oh ok so its \[\sqrt{2}e^\theta from 0-2\pi\]??

Answer 20

please confirm my final answer \[\sqrt{2}(e^2\theta-1)\]

Answer 21

you mean
\[ \sqrt{2}\left(e^{2\pi} -1\right) \]
about 755.885 numerically

Answer 22

yes thanks. got it now, can I ask another question here, or do i need to open another thread. thanks alot.

Answer 23

It's better to open a new thread.