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Numerade Educator

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Problem 20 Hard Difficulty

Find the exact length of the curve.

$ y = 1 - e^{-x} $ , $ 0 \le x \le 2 $

Answer

$\sqrt{2}-\sqrt{1+e^{-4}}+\frac{1}{2} \ln \left|\frac{1+\sqrt{1+e^{-4}}}{1-\sqrt{1+e^{-4}}}\right|-\frac{1}{2} \ln \left|\frac{1+\sqrt{2}}{1-\sqrt{2}}\right| \approx 2.22142$

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Video Transcript

Hey, it's Claire. So in your radio. So we're gonna find the length of the curve for why is one minus e to negative X excess between zero and two. So the derivative we get is e to the negative X and our length Argh! Link formula from 0 to 2 square root of one plus e to the negative two x d x We're gonna substitute you square for one plus e to the negative to X So when we substitute to you, do you equals negative to e to the negative two x t x d X equals negative. You do you over you square minus one. So this changes the limits of integration and they become from negative to square root of two. Excuse me to one plus e to the negative four you times negative. You do you Over. You square minus one. We go here, this becomes equal to square it to square root of one. Pull us me to the negative for use square, do you over one minus you square which is equal to square of two square root of one plus e to the negative four you square minus one plus one over one minus you square. And this splits into two. So it splits into you. Square minus one over one minus. You square from negative too. To one plus me to the negative for do you plus read too. One plus eat the negative for do you over one minus You square We simplify this. This becomes equal to negative. You too. One. Pull us Eat the negative for do you plus route to one plus b to the negative for do you over one minus You square will continue. Here comes negative rich too. One plus e to the negative for D u plus 1/2 rich too One plus eats the negative for one minus you plus one plus you over one minus you times one plus you Do you When we simplify this this is equal to negative route too square root of one plus eat the negative for do you less 1/2 square it of two square of one plus eat the negative for becomes one over When mine issue plus one over one plus you do you And this becomes you When we ent I drive it This becomes l on negative l end of one minus you. Class Ellen of one plus you. And when we find this, this is equal to square root of two minus square root of one plus eat negative for plus 1/2 L n of one plus square root of one. Pull us eats and negative four over one minus square root of one plus Eat the negative for minus 1/2 well and one plus square of two over one minus square root of two. This is equivalent to around 2.22 one for two.