What is the arclength of the polar curve #f(theta) = csc^2theta-cot^2theta # over #theta in [pi/3,pi/2] #?
Let's start off with the formula for the arc length of a polar curve whereby:
First of all, it's important to note that
Notice anything about this function? A trig identity perhaps?
Therefore, this function would simplify down to r = 1! This makes everything much easier!
Substituting everything into the formula we get:
Integrating a constant with respect to theta we get:
Now substitute the bounds in:
Therefore, the arclength of the polar curve is
Hopefully everything was clear and concise! If you have any questions, feel free to ask! :)