Consider a curve defined by the function \(r=f(θ)\), where \(θ\) is in radians and \(α≤θ≤β\). In rectangular coordinates, the arc length of a parameterized curve \((x(t),y(t))\) for \(a≤t≤b\) is given by:
\[ L = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \ dt \]
In order to adapt the arc length formula for a polar curve, we use the equations:
\[\begin{align} x &= r \cos(θ) = f(θ) \cos(θ) \\ y &= r \sin(θ) = f(θ) \sin(θ) \end{align}\]
The derivative of these is:
\[\begin{align} \frac{dx}{dθ} &= f'(θ) \cos(θ)-f(θ) \sin(θ) \\ \frac{dy}{dθ} &= f'(θ) \sin(θ) + f(θ) \cos(θ) \end{align}\]
In the definition of \(L\), we can replace \(t\) by \(θ\):
\[ L = \int_\alpha^\beta \sqrt{\left( \frac{dx}{dθ} \right)^2 + \left( \frac{dy}{dθ} \right)^2} \ dθ \]
We already know the definition of \(\frac{dx}{dθ}\) and \(\frac{dy}{dθ}\):
\[ L = \int_\alpha^\beta \sqrt{\left( f'(θ) \cos(θ)-f(θ) \sin(θ) \right)^2 + \left( f'(θ) \sin(θ) + f(θ) \cos(θ) \right)^2} \ dθ \]
Expanding and simplifying:
\[\begin{align} L &= \int_\alpha^\beta \sqrt{ (f'(θ))^2 (\cos^2(θ) + \sin^2(θ) ) + (f(θ))^2 (\sin^2(θ) + \cos^2(θ)) } \ dθ \\ &= \int_\alpha^\beta \sqrt{ (f'(θ))^2 + (f(θ))^2 } \ dθ \end{align}\]
Writing this in terms of \(r\):
\[L = \int_\alpha^\beta \sqrt{ r^2 + \left(\frac{dr}{dθ} \right)^2} \ dθ \]