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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?

  • 2
    Why find $f$? The formula for arc length involves only $\frac{dy}{dx}$2012-11-05

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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.