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Find the length of the following two-dimensional curve.T r(t) = (3cos 31, 3sin 34 , for Osts S 3...

Question

Find the length of the following two-dimensional curve.T r(t) = (3cos 31, 3sin 34 , for Osts S 3

Find the length of the following two-dimensional curve. T r(t) = (3cos 31, 3sin 34 , for Osts S 3



Answers

Find the are length of the given curve. $$r=3-3 \cos \theta$$

According to the caution, We need to find our planned. We have given equation. Is Apple Trudeau? Is it going to Oh, plus three. Sign 32 down. We know that we're using different Children whether it's back to Trudeau when we differentiating with respect to today we have a dash of Quetta duties. Nine. Ghost Rita. We know that the ark plant is is as his he could do the route have dashed through the whole square Bless half of Tuta Whole square did data. When we substitute this value in this equation, we have it even go square three cheetah plus to bless three signed three Judah whole square digital. When the further sold this equation, we have it even course square Trischitta Bless four. Plus there is a plus Behold scared Askar plus B squared plus 24 plus nine Sign square Trita nos love sign Credo Digital When we sold this equation be using integrate Garica Lido Men VI Use this calculator The value asks govern Saudis 41 point fight for one

According to the question, we have to find the art lend we have in given equation In the ocean is Abdou dough physical toe to cause treater. We know that in this are planned with differentiate with respect to data Then we differentiated with respect to theta. We have a dash of Kyoto, but we're different. Your dysfunction you have the value is minus six. Sign three Tito, do you know that? Therefore the are Clint is the are planned. He's as his equal do zero to Dubai route after dash to the whole square plus apple Tuta, hold scare digital the use The statement we pulled the value have dashed off the whole scare and 1/2 of Peter. We have zero to buh Bye. Road 36 Fine square. Trita plus four course square. Do you know? Did you know when we saw we use integrate Calculated? Oh, when we use integrate calculator, we have the value off ass is 13 point 36 Fight

Okay in this video we're going to look at the parametric equations X. Is equal to Um three co sign T minus coastline. Three teen and why is equal to three science? He minus sign through to you. And we're looking at the interval from 0 to Pi in T. And so what we would do in this question is find the arc length to find the length of the curve Between zero and Pi. And so we know the formula of arc length is just interval from time one to time to of the square root of the X. DT squared plus dy DT squared. And then this is all with respect to T. We can plug in our known balance. We know we're going from zero to pi and we're gonna have a lot of terms. So let's go ahead and draw big radical and start without derivatives. So we're looking at the X. Which is three. Co sign t minus three minus cosine three T. You can see that the derivative is just -3 -3 sci fi in the first term and then Positive three signed 3 t. in the second term. And then we're adding on the why derivative which is the rico science. He for the first term minus three co sign three teeth the second turn and this is all squared as well. And this is all with respect to T. 18. Okay so now we just have to solve this integral. Um but it's pretty easy if you know a lot of your trig identities. Let's go ahead and start writing um Notice how each of these terms has a three and they're all in squares. This means that we can factor out a three squared out of everything. And we should end up with this case. We'll write it in order will say to sign 30 minus scientist. That's weird. Plus coastlines. He minus coastline through to you squared. And this is all were expected to, This is three squared under the square root. We can pull it out as a. three and let's go ahead and expand these these uh these binomial Starting for the first term we have sine squared of three teeth and then adding on two signed three T. Sci fi. And then finally sign square T. And now with the second set we end up with co sign square to you minus two. Co Sign three T. Costa and see. And then Plus co sine squared of three teen. And now it easy. So we can see we have these matching sets of science squared and cosine square in the T. And then sine squared and cosine squared in three T. And so it doesn't matter what you plug in. If you have a sine squared plus cosine squared and it's the same Variable. It will equal one. And so we have to set so we have to and then we have minus two times. We can combine this quantity signed three t. 70 Plus Co Sign three T. Co sign. And this is all respected too. And just looking at this we can see that we can pull out a square root two. Let's start off with that three screw to Go from 0 to Pi. And so now we're left with one minus um this quantity Sign three t. sci fi and then minus co sign three T. Co sign T. Are respected too. And so this is now the integral that we have to solve. Um Will notice that the hard part is this stuff here these signs and co signs. But if we go ahead and remember a a trick identity. The angle edition especially will be very useful. So in this case ankle edition or in particularly angle subtraction we see the coastline of alpha minus beta is equal to cosign alpha to assign beta. My ah plus sine alpha sign beta. And you can see that this matches this except we have how far 0- three T. And beta is equal to T. So if we go ahead and plug this in we see we have this we still keep everything out front And we end up with one The co sign of three T- T. Which is the same as 1- the co sign up to T. GT. Now there are many different ways you can write a coast line of duty. One of them is the following. We have co sign of two T. Is equal to. In this case we have two. Um Let's see let me go ahead and write this. It's actually in this case it would be 1 -2 Science squared of T. And so what we can do is pull again. That's so we end up with one and then minus coast entities. We have minus one plus two sine squared of T DT. And you'll notice that this gives us a simpler answer. As europe I Of the Square Root of two. Science. Where to? It's here. We can go ahead and pull everything out. We end up with another skirt too, Leaving us with six and sometimes the integral of transited pie of Sign Tgt. And it's pretty easy from here we end up with negative coastline T. As the anti derivative and plugging in zero and pi we get six times In this case, one plus 1 Or 12 as the final answer for this integral. So it was a pretty complicated its role. But overall the arc length is just simply 12

According to the question, We have to find our cleanse. We have the given from Chinese Apple Duda is equal toe sign Trita We know that we differentiate dysfunction with respect to Duda with respect potato Then we differentiate this function with respect to theta We have have Daschle Dita you know that the definition off scientist is called strata We have two equals three cheetah on Then we differentiate this angle This angle is treater There is three the wrist Rezko's trita therefore dark lend do our land He's as his equal do zero to Dubai limit Rude after dashed it The whole square bolos have death to doubt was good plus half of you don whole scared digital. Then we subtitled these value in this equation we have as his equal toe zero to go by when receptor told the value off Abdullah Streeter Duri's sign Square cubed feta plus nine course square Fujita, Do you know when we saw the use integrate calculated Oh, Then we use integrate calculator. You know that the value off s is equal to tartine 0.36 fight


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