The Arc Length, L is given:
from parametric length equation we have the familiar:
s = int_a^b sqrt(((dx)/dt)^2 + ((dy)/dt)^2) dt it is just the aggregate distance formula that you may have seen many time. If you think the polar coordinate as a parametric representation in r, alpha then:
r = f(alpha)
x= rcosalpha=f(alpha) cosalpha
y=rsinalpha=f(alpha) sinalpha
[(dx)/(dalpha)]^2 = [f'(alpha)cosalpha - f(alpha) sin(alpha)]^2
[(dy)/(dalpha)]^2 = [f'(alpha)sinalpha + f(alpha) cos(alpha)]^2
Add and Simplify:
[(dx)/(dalpha)]^2+[(dy)/(dalpha)]^2= f'^" 2"(alpha) + f^2(alpha)
L = int_(alpha1)^(alpha2) sqrt(f'^" 2"(alpha) + f^2(alpha)) " "dalpha
Now evaluate:
f'^" 2"(alpha) =[ d/(dalpha)(cos(3alpha-pi/2) + alphacsc(-alpha))]^2
f^2(alpha) =[ cos(3alpha-pi/2) + alphacsc(-alpha)]^2
Integrand, I =sqrt( f'^" 2"(alpha) + f^2(alpha)); alpha = t
I =sqrt(3cos3t + ((t cott - 1)csct)^2 + (sin3t - tcsct)^2
L = int_(pi/12)^(pi/8)(I) dt
At this point you ca use an Online Integration Calculator and get
L ~~ .2037
