Question

How do I find the arc length of a spiral?

Answer 1

The spiral is given by the parametric equations x=e^-tcost and y=e^-tsint.

Answer 2

@agent0smith

Answer 3

@SithsAndGiggles

Answer 4

Oh and I need to find the length from 0 to infinity.

Answer 5

If I'm not mistaken, the arc length of a parametric path like this one is given by
\[L=\int_a^bdS=\int_0^\infty\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}~dt\]

Answer 6

And is that
\[\begin{cases}x=e^{-t}\cos t\\y=e^{-t}\sin t\end{cases}~~?\]
Or are the trig functions contained in the exponent?

Answer 7

Ok thats what I thought. It just seemed a little too simple to be right. lol

Answer 8

No what you have is correct.

Answer 9

Okay, so find the respective derivatives and plug it into the formula.

Answer 10

Ok thank you!

Answer 11

yw