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How to find the arc length of $f(x)$ on $\left[0,2\right]$which satisfy $$\frac{dy}{dx}=\sqrt{(1-x)^2+1}$$

I'm considering find $f(x)$ first and then calculate by arc length formula. However I don't know how to deal with the sqrt when integral.

Could anyone give a fast solution?

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Are you looking at length of $f(x)$ or length of $\frac{dy}{dx}$? –  B. S. Nov 5 '12 at 11:55
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It looks like maybe the problem is asking about the arclength of the function $y(x)$, and giving you the derivative $dy/dx$ already. Since only $dy/dx$ appears in the arclength formula, you can get the arclength only knowing $dy/dx$ without knowing the original function $y(x)$. –  coffeemath Nov 5 '12 at 11:57
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Why find $f$? The formula for arc length involves only $\frac{dy}{dx}$ –  PAD Nov 5 '12 at 11:58
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1 Answer

up vote 1 down vote accepted

If you want to evaluate the length of $f(x)$, so use $ L = \int_0^2 \sqrt{ 1 + (\frac{dy}{dx})^2 } dx $ which is $$ L = \int_0^2 \sqrt{ 2 + (1-x)^2 } dx $$ The above integral can be evaluated by taking $1-x=\sqrt{2}\tan(t)$. For a solid information see Intuition behind arc length formula and @Arturo's answer.

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