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Find the arc length parameter along the curve from the point where t = 0 by evaluating the integral $ lvldr. Then findthe length of the indicated portion of the cur...

Question

Find the arc length parameter along the curve from the point where t = 0 by evaluating the integral $ lvldr. Then findthe length of the indicated portion of the curve_r(t) = (e'cost)i+ (e' sint)i - e'kIn 4sts0The arc length parameter is s(t) -

Find the arc length parameter along the curve from the point where t = 0 by evaluating the integral $ lvldr. Then find the length of the indicated portion of the curve_ r(t) = (e'cost)i+ (e' sint)i - e'k In 4sts0 The arc length parameter is s(t) -



Answers

Parameterize the curve using the arc-length parameter $s, \quad$ at the point at which $t=0$ for $\mathbf{r}(t)=e^{t} \sin t \mathbf{i}+e^{t} \cos t \mathbf{j}$

Falling occur function and we're us to find it startling perimeter and then using that formula buying the length so the our plant perimeter can be found by taking the integral from sea to t o B. Tell detail might be in here is to speak if you notice I don't put the factor had is I'm getting here just not aiding that. That's a speed. So let's find the speed To find the spirit we take the driven er which is our velocity. So do be had equals the derivative of each component. So over here this is a product rural. So it will be the video will eat to t, which is just a to the tee times course I Inti Andrea, of course, I antes native scientists. There will be it to the t sign t hi. And then because again product rule e to the T Sainty plus again e to the T ho Scient E j and then degenerative off eternity. Just each of the tea, okay? And then, to find speed, we take the distance formula so it will be okay to the to t course. I'm skirt key minus two e to the t e to the t close eye anti ST E plus e to the to t sine squared t. And then same for J component up and then price key the key coast sine squared key on the last one. It will just be placed e typically. So this is really long, but something's cancel out so days will cancel out. And then if they actually paid enticing these and these thistles a trick identity. If you pull out to the tune T so it will just be factor will just equal to And I'll just not give me different colors. So it will just be e to the to to from there no one over here. This is another one again if you pull out e to the to t that's the same square post close aides were identity, which is just one. So, plus, each of the two t Ellen are less. One is this one. So cause e there it needs to be to eat to the 20. So there's just gives us Route three and discarded off into the tea is just okay. Todo t So this is our speed. This is what we need to take integral off so R s F t t from zero to t oh, Route three e to the towel. T tell. We'll just equal to Dane Trickle of e to tell is just see two to tell So we'll be rich free and retrace the constant So we're here. Olivia's far zero to t on. This is just regret. Three gate to the T minus E to the zero is one. So, Ruth three, this is our arc length permit. Er, now we can use out to find the length and to find the length you just play game are pulling for tea on our T will be native Ellen of four, so it will be hurt. Three key to the native Ellen a poor minus root. Three. This parting here is one force. So be Route 3/4. My name's Rick three. This gives us Natick three route three hour. For however, art length is always positive. So we looked his magnitude So disa logistical or three for three over four. And this is our answer for the second party

So we have a falling function of a Kurban. We are us to find our clients perimeter. And then using that, I find the length. So to find the arc length family dirt watches right to formula down. So a softy will be the integral from zero to T Aldo speed B o Tell De Vito and then to find speed, we just take the first derivative which is gives us our velocity. So our be prime are are trying which is back there b equals this is a product ruling here. So it will be eat, Ricky, cause I anti diarrhoeal coastline is negative saying I and then it will take to dread o of J, which is again a product rule. So plus lead to the T sank e plus did your it off Sinuses causing so into the team hosts anti okay and druid off e to the t just e to the t. Okay, so this is our but, alas, it er prime. And then we'll take its magnitude. And we used to sense formula for Dad. So our speed will be in here. It's e to the to t co sign squared cle minus two e to the T host Sainty silently plus a to the to T signed square T And then same for J plus E to the to t signed Square T and then cause to lead to Clea Sancti Co Sainty and then plus E to dio to t closing square key. And for the last one is just because he to the to t this is really long. But we can cancel out this point this and then over here we actually have treek identities, which makes our work much nicer and easier. So over here, this is one if you pick out each of the two t biggest science where postcards Times Square, which gives us one. So I'm just gonna break that with different colors. So this would be one a charity because and there's another one over here. So this is another each of the two t and then we had the last one US e to the two teams. Things gives us Route three to the two t. We can take out e to the to T. So will be Route three t to the T. So this is our speed on when we can use it to find our our plane. So our art line is s o At T equals zero to tee off. Brute three be Tell Deke tell So rotaries a constant and for integral into the Tao is just eat two to tell. So we have Rick three into the tell you violated from zero to t this whole just equal to brut three e to the t e t zero is why. So it will just be minus root three. So this is our art length perimeter. Now we can use it to find the length and then for the length we had tickles. Negative, Eleanor four. So it will be as self negative I leno for well, a gold too great. Three d to the native Lono for minus root three. Thes over here is just one force, so we have as follows minus your three. If they combine, we get negative drink Route three over four. However, aren't length is always positive because its length quantity. So we take the magnitude of it when we get three. Route 3/4. And this is our second part of the question

So here we're trying to find the arc length of the curve. Um So we're going to be using the formula for arc length. And that is going to be the integral from alpha to beta or from A to B. Of the square root of mm. The partial derivative X. If you expect the data squared plus the derivative of Y. With respect to theta squared the fatah. So like this in mind. Um we find the XD data and the wider data based on what we're given. Um So our arc length formula since we know that the Y. D. Theta is E tha tha coastline data plus sign data. E. To the data. We end up getting that our art length um is going to be the integral from high over to pine of route, chew E. To the paper. D. Theta. And that's going to get this value right here. Which is the same thing as root two times E. To the pie minus E. Yeah, hi over chip Latinos are the same values.


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