f-x-5-687cosh-x-5-687-5-687-L-11-11-1-f-x-2-dx-

Question Number 40906 by Penguin last updated on 29/Jul/18

f (x) = 5 . 687 \cosh (\frac{x}{5.687}) - 5.687

L = \int_{- 11}^{11} \sqrt{1 + {[f^{'} (x)]}^{2}} d x

Commented by maxmathsup by imad last updated on 29/Jul/18

f (x) = λ c h (\frac{x}{λ}) - λ w i t h λ = 5, 687 \Rightarrow f^{^{'}} (x) = s h (\frac{x}{λ}) \Rightarrow

L = \int_{- 11}^{11} \sqrt{1 + {s h}^{2} (\frac{x}{λ})} d x = \int_{- 11}^{11} c h (\frac{x}{λ}) d x =_{\frac{x}{λ} = t} \int_{- \frac{11}{λ}}^{\frac{11}{λ}} c h (t) λ d t

= 2 λ \int_{0}^{\frac{11}{λ}} c h (t) d t = 2 λ {[s h (t)]}_{0}^{\frac{11}{λ}} = 2 λ s h (\frac{11}{λ}) = 2 λ \frac{e^{\frac{11}{λ}} - e^{- \frac{11}{λ}}}{2}

= 5, 687 (e^{\frac{11}{5, 687}} - e^{- \frac{11}{5, 687}}) .

Answered by tanmay.chaudhury50@gmail.com last updated on 29/Jul/18

f (x) = a c o s h (\frac{x}{a}) - a

f (x) = a \frac{e^{\frac{x}{a}} + e^{\frac{- x}{a}}}{2} - a

f^{'} (x) = \frac{a}{2} (e^{\frac{x}{a}} . \frac{1}{a} + e^{\frac{- x}{a}} . - \frac{1}{a})

f^{'} (x) = \frac{e^{\frac{x}{a}} - e^{\frac{- x}{a}}}{2}

{[f^{'} (x)]}^{2} = \frac{e^{\frac{2 x}{a}} + e^{\frac{- 2 x}{a}} - 2}{4}

1 + {[f^{'} (x)]}^{2} = {(\frac{e^{\frac{x}{a}} + e^{\frac{- x}{a}}}{2})}^{2}

L = \int_{- 11}^{11} \frac{e^{\frac{x}{a}} + e^{\frac{- x}{a}}}{2} d x

= \frac{1}{2} ∣ e^{\frac{x}{a}} . a - e^{\frac{- x}{a}} . a ∣_{- 11}^{11}

= \frac{a}{2} {(e^{\frac{11}{a}} - e^{\frac{- 11}{a}}) - (e^{\frac{- 11}{a}} - e^{\frac{11}{a}})}

= a . (e^{\frac{11}{a}} - e^{\frac{- 11}{a}})

= 5.687 (e^{\frac{11}{5.687}} - e^{\frac{- 11}{5.687}})

p l s c h e c k \dots

Answered by MJS last updated on 29/Jul/18

\frac{d}{d x} [a \cosh \frac{x}{a} - a] = \sinh \frac{x}{a}

\sqrt{1 + \sinh \frac{x}{a}} = \cosh \frac{x}{a}

\int \cosh \frac{x}{a} d x = a \sinh \frac{x}{a}

\Rightarrow L \approx 38.525

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