What is the arclength of $f (x) = x - \sqrt{e^{x} - 2 ln x}$ $f(x)=x-sqrt(e^x-2lnx)$ on $x \in [1, 2]$ $x in [1,2]$ ?

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What is the arclength of $f (x) = x - \sqrt{e^{x} - 2 ln x}$ $f(x)=x-sqrt(e^x-2lnx)$ on $x \in [1, 2]$ $x in [1,2]$ ?

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Answer 1 · 2017-03-03T04:10:32

$L = \int_{1}^{2} \sqrt{1 + {(1 - \frac{x e^{x} - 2}{2 x \sqrt{e^{x} - 2 ln x}})}^{2}} d x \approx 1.0630$ $L=int_1^2sqrt(1+(1-(xe^x-2)/(2xsqrt(e^x-2lnx)))^2)dxapprox1.0630$

Explanation:

The arc length of the curve of $f$ $f$ on $x \in [a, b]$ $x in [a,b]$ is given by

$L=int_a^bsqrt(1+(f'(x))^2)dx$

Here, $f(x)=x-(e^x-2lnx)^(1/2)$ so

$f'(x)=1-1/2(e^x-2lnx)^(-1/2)(e^x-2/x)$

$color(white)(f'(x))=1-(xe^x-2)/(2xsqrt(e^x-2lnx))$

Then the arc length is given by

$L=int_1^2sqrt(1+(1-(xe^x-2)/(2xsqrt(e^x-2lnx)))^2)dxapprox1.0630$

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What is the arclength of $f (x) = x - \sqrt{e^{x} - 2 ln x}$ $f(x)=x-sqrt(e^x-2lnx)$ on $x \in [1, 2]$ $x in [1,2]$ ?

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What is the arclength of $f (x) = x - \sqrt{e^{x} - 2 ln x}$ $f(x)=x-sqrt(e^x-2lnx)$ on $x \in [1, 2]$ $x in [1,2]$ ?