Concept:

Arc length or curve length is the distance between two points along the section of the curve. Determining the length of an irregular section of the arc is termed as rectification of the curve.

The length of the curve y = f(x) from x = a to x = b is given as:

\(l= \mathop \smallint \limits_{{\rm{x}} = {\rm{a}}}^{{\rm{x}} = {\rm{b}}} \sqrt {1 + {{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)}^2}} {\rm{dx}}\)

Calculation:

Given curve is y = log sec x

\(\frac{{dy}}{{dx}} = \frac{1}{{\sec x}} \cdot \sec x\tan x = \tan x\)

Hence the required length will be:

\(S = \mathop \smallint \nolimits_{x = 0}^{\pi /6} \sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}dx} \)

= \(\mathop \smallint \nolimits_0^{\pi /6} \sqrt {1 + {{\tan }^2}x} \;dx\)

= \(\mathop \smallint \nolimits_0^{\pi /6} \sec x\;dx = \left[ {\log \left( {\tan x + \sec x} \right)} \right]_0^{\pi /6}\)

= \(\log \left( {\frac{1}{{\sqrt 3 }} + \frac{2}{{\sqrt 3 }}} \right) - \log 1 = \log \sqrt 3 \)

= \(\frac{1}{2}\log 3\)

Important Note:

If the curve is parametrized in the form x = f(t) and y = g(t) with the parameter t going from a to b then

\(l = \mathop \smallint \limits_{{\rm{t}} = {\rm{a}}}^{{\rm{t}} = {\rm{b}}} \sqrt {{{\left( {\frac{{{\rm{dx}}}}{{{\rm{dt}}}}} \right)}^2} + {{\left( {\frac{{{\rm{dy}}}}{{{\rm{dt}}}}} \right)}^2}} {\rm{dt\;}}\)