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Find the length of the arc of the curve from point $ P $ to point $ Q $.

$ y = \frac{1}{2} x^2 $ , $ P(-1, \frac{1}{2}) $ , $ Q(1, \frac{1}{2}) $

$L=\sqrt{2}+\ln (\sqrt{2}+1)$

Applications of Integration

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it's clear. So when you read here So we're gonna find the length of the curve of why is equal to 1/2 x square X is between negative one and the one included. So we have dealing to be equal to negative 11 square root of one plus x squared e x When we're gonna find the integral we're gonna substitute you. So he got we just cross out negative one and one for a second and just focus on the square root part of it. You get one plus you square, we'll make you be equal to tangent data. So do you seek it square D data. So this is equal to the interval of seeking cute these data. So we're gonna use integration. My parts and we get seeking sequence square E data minus E of seeking data over E data seeker squared. He data this becomes equal to seek it. Data guns tangent beta minus into a girl of seek it. Times 10 gin times 10 Gin de Saito We'll continue here, which is equal to seek it. Times 10 gin minus into girl of speaking times. Tangent Square Ms becomes equal to seek it times tangent minus tentacle of seeking cute minus C Can't data money rearranged This we get to integral of seeking cubed data is equal to seek it times tangent plus l And, um, second plus tangent plus c We just moved up to and divide everything like to So we get into bro. A second cute of beta is equal to 1/2 Seek unt una times tangent data plus one have. Oh, and oh, second plus tangent plus c, We're gonna substitute feta to be equal to art. 10 Geant This gives us you over to square it, um you square plus one plus 1/2 elf in of you plus square root of one plus you square plus c. We're gonna find this from negative 1 to 1. But we don't need the C. When we calculate this, we get rich too. Plus oh, and of route two plus one