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Problem 10 Medium Difficulty

Evaluate the line integral.
$\int_{C} 3 y^{2} d y,$ where $C$ is the quarter-circle $x^{2}+y^{2}=4$ from (0,2) to (-2,0)

Answer

$=24 \int_{\pi / 2}^{\pi} \sin ^{2} t \cos t d t=-8$

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Video Transcript

given the integral three violence for Divi only Cove X squared plus y squared is to go to full on Gardiner Za 0/2 to minus two, comma zero. So if I substitute by is equal to two scientific somewhere, the viable big will too to force t didn t So my stink of three vice where the very will become three into to scientific Old Square into do course d did he? So this is it will do three into four science where d into to costea de de So we get reports that will well into two is 24 multiply by Costea sine squared d de t Now if it's a scientist, is he good to you? We get the you as course d do you for therefore that a substitute? And we know that the tea is changing with the zeal is an equal to teal as an equal to pi Because according there's are from 0 to 2, then minus 2 to 0 But this will be zero to buy So zero to buy sign is you square do so in general of you squared u is u to the power three by three from zero do by. So this is it Will do. Eight. You was signed to t 0 to 5. You know that scientists zero had both devalue spying zero diplomat integral value is zero.

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