5

Find the exact length of the helix 7(t) -5 < t < 2.8 cos4), 8 sin3t for...

Question

Find the exact length of the helix 7(t) -5 < t < 2.8 cos4), 8 sin3t for

Find the exact length of the helix 7(t) -5 < t < 2. 8 cos 4), 8 sin 3t for



Answers

Find the length of the curve.

$ r(t) = \langle t, 3 \cos t, 3 \sin t \rangle $ , $ -5 \le t \le 5 $

Even the rt you could you 1/2 ghosts. I'm 31 half some dirty and the square root on the tree out of 40 on Dhere. We need to find our land. I'm going tone. So one turn here. So one turn were given to the one complete physical. And because the discourse and this I will be the beard I'm equal Jew the period pickled geo job I Therefore, we innocently blast that they will be between Georgia to pie will become it one turn here and we know the Atlantic. Ask me, Kiko Junuh, Integral from a to B the gnome under arm from D the day here we are given the and be already here and we need to find a Bram. So let's and you find a broom here for us and then we get Echo Drew the a man Us Ah ah! Half side day and 1/2 Go side day This one will be the square of military And too far And now been no Mom the Bronte and we go Jew the square root off the minus 1/2 psy d square Plus half course I they square plus let three out of four and we can signify this one. And then we get equal to the square root we build. Ah, half sky. Also we wanted for our side and then we have this. I squared the press. Of course I squared the plus three out of four. And because it's an echo to one, therefore we have discovered on the one out of far less trade of far. And it could just go to one and equal to one. They're fun out the our coach you into car from Sergio Troop I The number at Bronte go to one. Did they? They're fortunate. Could you t He violated from 0 to 2 pi and it could YouTube prime. And it's ruined. It could YouTube by and that would be the answer.

Okay, so we're asked to find our clink of Margie, which is equal to to sign of tea are empty and Yuko sanity. And this is from T is between negative 10 2 10 case that we have the interval from negative 10. You're 10. Well, what do we need to do? First, we need to take the derivative of our privacy for Artie. So we get primitive is to co sign of t comma five comma. Negative to sign. Lt. Okay, now we need the magnitude of that. Does that give me that? This square bit of for coastline party plus 25 plus for science 40. Well, for a factor, the four from Close and Corti put signs, court scene. We see that question is where to put signs where she is just one. So there's this four plus 25 that's just square roots of 29. Okay, so let's go to 29. That's this is going to 29 d f d. And we get spirit of 29 times t evaluated at 10 in the person and that gives me the square root of 29 10 minus. Make up 10. So our solution is just 20 scourge of tony night.

To find the or cling. We'll begin by identifying that we have three dimensions here an ex a Y and a Z dimension. And this is Ah, Parametric equation. And so we're going to set up the integral from 0 to 4 pi of the square root of DX DT, which is going to be the derivative before, which is one and then square. This is Ah, Pythagorean theorem and four dimensions is the derivation of it. Which is why it looks like this, plus the derivative of eight Cy Inti which would give us eight coats I, Inti all squared, plus the derivative of a co sign T which would be negative. Eight sign Teoh all squared. Okay, so from here, we're actually going to be able to use some simplification toe help us a bit because it looks a little messy right now. But that's OK. It's gonna be much better in second. So we still have being a girl from 0 to 4 pi or and we have square of one. Who else at this point, what we're gonna do is notice that there's going to be a term of 64 here eight square and the term of year also posited 64 8 squared. And so we could on factor both of those out and say, Swear about 64 parentheses. And then we have our consign squared t and bus our science bird, and then we can use the Trigana metric identity. That science bird plus coat sine squared is just going to be one. And so all of that will go ahead and simplify the one. And so we're really just left with the square root of 65. And so from here on, it's just another stuff or to toe take the anti derivative of that so we could set up. We now are going to take the integral of spirit of 65 will go and write this next step out here. But since it the constant on the anti derivative of it, is just going to be square. The 65 t So you square me 65 to evaluated from or pie zero or 0245 and then finally plugging in our upper limit. First subtracting are lower limit. We'll get a final answer of or pie square Room 65 and this will be the Ark link or this problem, and that's it

Hey, guys. So given a three dimensional per, um, given his rt and that we're gonna be using this equation in order to tell us the are playing. So firstly, we should find our prime Just the cool to the derivative respect t of our d So we do. That would be This piece is we get three of them. All right, So when I take the derivative of tea, right, that's just gonna give us one. I take the derivative of eight sine of t. Here, take the derivative of that. That's just gonna be eight times the derivative respected TF side of T, which is just a co sign of tea. And then same kind of thing when a Jake the dream of a co sign of t respect T That's just gonna give us a negative sign of T heir. Sorry. Negative. Eight of tea. Oops. Make that pretty, You guys. Uh and so then this is our our prime tea. Now in each need to find the magnitude which isn't too. So that's gonna be is the square root of each of these components square. So then we're gonna get one squared because t squared, plus negative. It's enough t squared. And then what we do that we get something? Not too bad. All right. We're just gonna get the square root of one. Right? Because one squared, which is one plus eight times or eight co signed a T squared, which isn't gonna be 64 co son squared t plus 64 sign swearing of tea. And then we can do is we can factor the 64 out from those terms and we can use our Sagara and identity. It's a four. Sorry. It was science creative T. Plus I squared of tea. That, of course, this right here is equal to one. So then we just get this square root, uh, 65 which is not a perfect square number, but that's fine. Now, we're just going to enter this into our inner Groll from 0 to 4 pi square with the, uh, 65 e t. Just then just going to be screw a 65 t innovative 0 to 4 pi six. Sorry, it's definitely a zero. And from there, that's just gonna give us four pi square of 65. That would be or given or clicks


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