Find the length of the exponential spiral y = e^ -(theta)?

Question

from r = 1 to the origin

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this is polar coordinate
theta is the angle

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TYPHO GUYS!!

r=e^(-theta)

Answer 1

The general formula for an exponential or logartithmic spiral is given by:

r = a*e^(b*theta)

For your case, 'r' is the same as 'y', and (a = 1) & (b = -1)

r=1 implies:

1 = e^(-theta)

this is only possible when theta = 0.

The arc length of your exponential spiral (from r = some value, to the origin) is given by:

s = (a/b)*sqrt(1+b^2)*e^(b*theta)

Hence, for your problem:
s = (1/-1)*sqrt(1+(-1^2))*e^(-0)
s = -1*sqrt(2)
s = -1.414

Hence, according to my calculations, the arc length or distance is equal to sqrt(2), or roughly 1.414 units.

P.S. I think the negative sign indicates that it is a clockwise spiral

Answer 2

Spiral length from r=1 to origin:
L = 1

derivation:

let t = theta in radians
Construct a differential length element dL based on differential angle dt
dL = r*sin(dt)
r = exp(-t)
sin(dt) = dt

hence
dL = exp(-t)dt
r = 1 = exp(0) : t = 0
origin = 0 = exp(-infinity) : t = infinity

and therefore
L = integral(0,infinity, exp(-t)dt)
= [-exp(-t)]{0 -> infinity}
= [-exp(-infinity) + exp(0)]
= [ 0 + 1]
= 1

Answer 3

What would the number theta be?

Answer 4

Take a look at:

http://mathworld.wolfram.com/LogarithmicSpiral.html