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Problem 39 Hard Difficulty

Find the exact length of the curve.
$$x=t \sin t, \quad y=t \cos t, \quad 0 \leqslant t \leqslant 1$$

Answer

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

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Video Transcript

all right, so this probably were again looking for the length of the curve. And we know that our length will be equal to ventricle from 0 to 1 of the square root of my DX dy t squared plus my d y did he squirt? So first, let's take the derivative of X with respective feet. So I have to use the product rule that they're using the product rule. I'll get sign it, see plus t Times CO sign of tea. Likewise, when I take the derivative of why with his back to tea, I'll have to use the product roll thing. I'll get co sign of tea minus t sign it, See? And this minus is coming from the fact that the derivative of co sign is negative sign. Okay, so now with this, I can't square everything, and I'm gonna work on what's inside the integral separately for a little bit. So when I scared square my green apart, I will get sine squared of teach plus t sci fi co signer Teeth plus XI squared times close sine squared Lefty. That will be plus my Red Part Squared, which is co sign scored a T minus T sanity Po Sciences plus C squared who sine squared of tea. So some things to know my middle terms will cancel out with each other. And then I will be left with sine squared a T plus coastline squared of teeth. Plus my last two terms both have c squared. So all factored out from them to be left with close science quite at sea plus science Gordon C free. So notice I have science court t plus co star Discordance e and I have it again. We know with trig identities that both of these will be equal to one. So because of that, my integral for my length reduces rather nicely. I'm simply left with square root of one plus t square, didn't she? But now, even though this is how much nicer a integral we still have to dio fix substitution. Okay, So if that we're gonna let t we need full to Tanja of data, Which means can I take the derivative of tea with respective data that will be secret squared off data So my d t is equal to seek it squared of data notice when t is equal to one. When fit. Uh, well, we'll choose the data to be hi divided by four. And once he is equal to zero, I say there will also be zero. So with those substitution Zinn's, we'll get that. The length of my curve is now from zero to pi over four of one plus tan square data times C K squared state a beef data. That's drop that over here. Can't forget that. Okay, now we know a trick. Identity. Specifically, we know that one plus tan square data is equal to Seacon Square Trader. So if I were just for rueful size, I would end up getting just the secret there. So the length of my curves was now equal to seconds later. Time sequence, squared data. Do you think we could combine these terms into secret cute data, however, and order to solve the integral we need to leave them like this. This is because we're now gonna use integration by parts to remind yourself this is when ventricle of you TV is equal to UV minus integral of the Do you. So I'm gonna let my you be seeking data and my devi be seeking square data. So for that my in trickle l will be you, which is seeking data. The anti derivative of Secret Square data is tangent data. I think I am going to have to valley that zero for minus then your girl. Well, the derivative, the derivatives. Sorry. The derivative of you is secret Data Tangent data. And then I will multiply by V, which we know V is tangent data, which gets me escorted there. I mean, my bounce stay the same. Thank. So now I'm gonna focus well for simplicity. Slate, Let's, um let's focus on this part really quick, so that part will be seek it of pi over four times tangent empire before finest ticket zero times tangent. So if we plugged that into our calculator knowing that seek it is one over coastline, we'll get that. This is equal to one two over the square root of two. So now our length is equal to two all over the square root of two minus center. Pull fe zero pile before change. Secret data. We're gonna use, uh, on identity to replace this tan square data. Alright. Tan scored later. Is seeking squares. Data minus one. July link is to over square root of two minus and a girlfriend. Zero high over four of seek it. Cube data. My mistake. It data. All of that is defender. So one thing is I'm gonna do is I'm gonna separate those into girls. She was this length vehicle to two over the square root of two minus zeros. Pile of pork. Seek it. Cute. Greater de Seda first integral from zero pi over four of sequence. Stater de Fainter. Noticed. This negative is now a positive because I have to distribute this negative out in the front. Thank so one thing to notice if we look back for been a girl of seeking cube off data looking right here, that's the integral of seeking cute. So that's actually the length of our curve. L So I'm going to substitute that in. So this is l equals two over the square root of two minus. That is, in fact, L plus zero the pilot before. Okay, So if I add lt both sides, I will get to our equals two over the square root of two minus heroes. Pyro for or let's actually find the anti derivative of that anti derivative of secret data. is the natural log on the absolute value of sequence data plus Tancredo. All of that will have to be evaluated. Zero in pi over four. Okay, so with that E, I have to l equals two over the square root of two minus. Have to be careful here. This is the natural log of Secret of Pi. Over 40 plus tangent of file before minus the national law. A secret of zero plus tangent of zero. So to our equal to open square root of two minus, I'm gonna stretch distribute at my negative here Well seeking of pyro for something we already know it is too over the square root of two tangent of pile before is one. Now that negative becomes a positive And I have the natural log co sign a zero or seeking of zero is one and tangent of zero is there So just the natural walk of one me so less first multiply this whole side by 1/2 so I can multiply this by 1/2 So I get the length distributing that 1/2 I know how gone Over the square root of two minus 1/2 the natural log This is already positive, so I do not have to worry about it. And the natural log of one is zero. So this part disappears. So this is my final answer for the length of the curve.

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