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

$$f\left(x\right)=5.687\cosh \left(\frac{x}{5.687}\right)-5.687$$$$L={\int }_{-11}^{11}\sqrt{1+{\left[f{}^{\prime }\left(x\right)\right]}^2}dx$$

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

$$f\left(x\right)=\lambda ch\left(\frac{x}{\lambda }\right)-\lambda with\lambda =5,687\Rightarrow f^{{}^{\prime }}\left(x\right)=sh\left(\frac{x}{\lambda }\right)\Rightarrow$$$$L={\int }_{-11}^{11}\sqrt{1+{sh}^2\left(\frac{x}{\lambda }\right)}dx={\int }_{-11}^{11}ch\left(\frac{x}{\lambda }\right)dx{=}_{\frac{x}{\lambda }=t}{\int }_{-\frac{11}{\lambda }}^{\frac{11}{\lambda }}ch\left(t\right)\lambda dt$$$$=2\lambda {\int }_0^{\frac{11}{\lambda }}ch\left(t\right)dt=2\lambda {\left[sh\left(t\right)\right]}_0^{\frac{11}{\lambda }}=2\lambda sh\left(\frac{11}{\lambda }\right)=2\lambda \frac{e^{\frac{11}{\lambda }}-e^{-\frac{11}{\lambda }}}{2}$$$$=5,687(e^{\frac{11}{5,687}}-e^{-\frac{11}{5,687}}).$$

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

$$f\left(x\right)=acosh\left(\frac{x}{a}\right)-a$$$$f\left(x\right)=a\frac{e^{\frac{x}{a}}+e^{\frac{-x}{a}}}{2}-a$$$$f^{\prime }\left(x\right)=\frac{a}{2}(e^{\frac{x}{a}}.\frac{1}{a}+e^{\frac{-x}{a}}.-\frac{1}{a})$$$$f^{\prime }\left(x\right)=\frac{e^{\frac{x}{a}}-e^{\frac{-x}{a}}}{2}$$$${\left[f^{\prime }\left(x\right)\right]}^2=\frac{e^{\frac{2x}{a}}+e^{\frac{-2x}{a}}-2}{4}$$$$1+{\left[f^{\prime }\left(x\right)\right]}^2={\left(\frac{e^{\frac{x}{a}}+e^{\frac{-x}{a}}}{2}\right)}^2$$$$L={\int }_{-11}^{11}\frac{e^{\frac{x}{a}}+e^{\frac{-x}{a}}}{2}dx$$$$=\frac{1}{2}\mid e^{\frac{x}{a}}.a-e^{\frac{-x}{a}}.a{\mid }_{-11}^{11}$$$$=\frac{a}{2}\{(e^{\frac{11}{a}}-e^{\frac{-11}{a}})-(e^{\frac{-11}{a}}-e^{\frac{11}{a}})\}$$$$=a.(e^{\frac{11}{a}}-e^{\frac{-11}{a}})$$$$=5.687(e^{\frac{11}{5.687}}-e^{\frac{-11}{5.687}})$$$$plscheck...$$

Answered by MJS last updated on 29/Jul/18

$$\frac{d}{dx}[a\cosh \frac{x}{a}-a]=\sinh \frac{x}{a}$$$$\sqrt{1+\sinh \frac{x}{a}}=\cosh \frac{x}{a}$$$$\int \cosh \frac{x}{a}dx=a\sinh \frac{x}{a}$$$$\Rightarrow L\approx 38.525$$

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