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Numerade Educator

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Problem 35 Hard Difficulty

Find the arc length function for the curve $ y = 2x^{\frac{3}{2}} $ with starting point $ P_0 (1, 2) $.

Answer

$\frac{2}{27}\left[(1+9 x)^{3 / 2}-10 \sqrt{10}\right]$

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Video Transcript

it's clear. So when you read here so we have f of facts is equal to two X to the three house power. So that gives us a derivative. The three ex So 1/2 power one plus that they're a bit of square. Gives us one plus nine x someone we plug it in to our bowling formula. We're just gonna change acts to t you get the integral from one to x square of one plus 90 DT, we're gonna make you be equal to one plus 90. So nine e t is equal to do you so d t is equal to do you divide it by nine. This changes the limits of integration. So it goes from one to arks and it becomes ton. Since we're adding nine and one plus nine x you of 1/2 power, do you over nine. This is equal to one over nine turns 2/3 power 2/3 comes you to the 3/2 power. This is from 10 to one plus nine x. This gives us two times one plus nine x the three house power minus 20 square root of 10 over 27