Question

Find the length of the arc of the curve y = log sec x from x = 0 to x =​

Answer 1

The length of the arc of the curve y = log sec x from x=0 to x=π/3 is log (2+$\sqrt{3}$).

Given

To find the length of the arc of the curve, y = log sec x from x=0 to x=π/3.

The length of the curve, y = log sec x              

                                        y = log sec x  -----> (1)        

Differentiating equation (1), with respect to x, which gives

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

                                            = tan x

Where,                            $(\frac{ds}{dx})^2$ = 1 + $(\frac{dy}{dx} )^2$

                                                = 1 + tan² x                      

                                                = sec² x

Arc length "S" of the given curve is calculated from x=0 in the direction of x increasing,

                                         $\frac{ds}{dx}$ = sec x

                                         ds = sec x . dx

Hence, $S_{1}$ denotes the arc length from x=0 to x=π/3,

                           $\begin{equation}\int_{0}^{S_{1}} d s=\int_{0}^{\pi / 2} \sec x d x=[\log (\sec x+\tan x)]_{0}^{\pi / 3}\end$

                                 $\begin{equation}S_{1}=\left[\log \left(\sec \frac{\pi}{3}+\tan \frac{\pi}{3}\right)-\log 1\right]\end$

                                      = log (2+$\sqrt{3}$) - 0        (log 1 = 0)

                                      = log (2+$\sqrt{3}$).

Therefor, curve y= log sec x = log (2+$\sqrt{3}$).

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