Question

Finding the arc length of Archimedean spiral.

Answer 1

Arc length ds = \[\sqrt{{r ^{2}+\left( \frac{ dr }{ d \theta } \right)}}d \theta = a \sqrt{\theta ^{2}+1}d \theta \] using \[r(\theta) = a\]

Answer 2

\[r(\theta)=a \theta \]

Answer 3

From 4000 pi to 4002 Pi:
http://www.wolframalpha.com/input/?i=integrate++%28x^2+%2B+1%29^%281%2F2%29++dx+from+4000pi+to+4002pi

However, I checked with a circle at (4002/2) = 2001 radius, but come up with pi*2*2001 = 12572 which is basically close to the answer I got from the spiral from this application:
http://www.giangrandi.ch/soft/spiral/spiral.shtml

So, how can the arc length of a spiral from 4000pi to 4002pi be about 72k length while the circumference of a 2001 radius circle be about 12k?

Answer 4

@IrishBoy123 you helped me with prior spiral problem, will you continue?

Answer 5

http://www.wolframalpha.com/input/?i=integrate++%28%280%2B%281.25%2F%282*pi%29*x%29%29^2+%2B+%281.25%2F%282*pi%29%29^%282%29%29^%281%2F2%29+x+sinx+*%281.25%2F2%2Fpi%29+dx+from+0+to+4000pi+%2F+integrate++%28%280%2B%281.25%2F%282*pi%29*x%29%29^2+%2B+%281.25%2F%282*pi%29%29^%282%29%29^%281%2F2%29++dx+from+0+to+4000pi+

Answer 6

change sinx to cosx to get x coordinate. Change 1.5 to whatever your space is in your branch between spirals. Change 0 for the starting point of the spiral. The spiral converges (0,-0.399)

@IrishBoy123 It don't converge to (0,-2), but rather (0,-0.4) HUGE difference. Please check if I am right... I updated the link above.