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Finding Arc Length, & Unit Tangent Vector, given a position function r(t)

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Transcript

Hello, Tom for everystepcalculus.com and everystepphysics.com this is a another nonsense problem by calculus said to me by a student. I would imagine it never appear out of test but I’m going to do it anyways regarding an arc. Arc length and the unit tangent vector okay so, let’s do it index 8 to get to my menu we’re going to scroll down here to arc length in RTC. The RT here that’s a position vector, a vector has magnitude and direction in this case its tangent. So, therefore whatever curve this turns out to be on a graphing which I don’t know how to graph it. But somebody made this function here some mathematician or person made this function and they’re trying to convince us that this solves stuff in life and I say it doesn’t I say it comes up with a number which will show you this number right now and here it comes up with units. As if that was important to anything and it’s also one numbers like taking six times seven coming up with forty-two and saying hey there you see it’s that’s the orbit around Neptune, Neptune or something like that you know some wild statement.

So, anyways let’s do it arc length and RT, we’re going to choose number 3 RT I have program the X and Y values also and I’ve already entered these to save time okay. All these functions but you’re gonna press alpha and put every one of these variables in here okay exactly like you see it, make sure you put time sign between the T etc. But I’ve already added them so here they are two times T times cosine not this right here I guess what the original function this one here okay and – T etc. So, I always show you, you can change it if you want I say it’s okay and we’re gonna do number four arc length okay. You can do all these things with what you’ve entered here. We’re gonna do and here’s the formula you write this stuff down in your paper for your test or homework or whatever. L is length of an arc it’s really a definite integral and that it uses the magnitude of the derivative of the RT function to come up with this stuff here and I show you all this you just keep writing in your paper putting it down you look like a genius. Now X of T is equal to remember I is X, J is Y, K is Z okay those are the parameters that they give us. So, the derivative of X of T this one right here is this right here two times cosine of T I’m not going to go through each one of it because it takes too much time for this video and here’s why T the next one – right here –  2 times T sine of T. Derivative of that is this right here of course you knew that didn’t you, you didn’t pull that stuff out of the air just like anybody can. Here Z of T here’s this right here and that turns out to the derivative is 2 times the square root of 2 times T okay.

Now, we’re going to find the magnitude we have to square all this stuff and here we have the integral here over A and B that range and we’re going to put in the range okay. So, I haven’t program that’s where we’re going to go alpha 0 for the value of T in the upper range is alpha second I. Okay, so I show you this is the integral we’re doing with regard to T I just put that in there in case you made a mistake on the range. So, X prime of T is equal to this and then we’re going to so then X prime of T2 is equal to this right here. Okay, the same thing with YT here’s the square of it and here’s the square of the Z parameter K. So, now we’re gonna do it over the range we got this that all in here we’re still doing the integral of this all stuff okay. You notice we got the square root of this here okay square root of this remember when you do an integral of the square root you change it to half, your exponent half right here. So, here’s what we’re still completing and of course the integral then is 2 plus 1 times the absolute depth T plus 1 okay and at T is 0 the problem as you put zero in for all the T’s and the answer for the integral. We get one and for pi we get PI plus 12 upper – lower always when you’re doing these this with a range a definite integral equals here right here so this is the answer pi squared plus 2pi.

Now, this is the answer that they get here on the right, right they giving you the answer okay so it’s 16.513 units. Now, what does that mean I mean you know it doesn’t mean anything to me doesn’t mean anything to you. It’s completely irrelevant you’re not gonna use this to go into space with let me tell you okay. So, this is the length of this unit tangent mention vector okay now we’re going to do more calculation we’re going to go back here number one and we’re going to scroll down here too unit tangent vector. You know tangible is given I’m against teaching this stuff but I’ll give you a little hint of what you’re talking about here and of course the length is one in a unit tangent vector okay. You can read that in your spare time and here’s the formula unit tangent vector is written just like this and all the calculus books derivative RT is over the magnitude of it and I show you what you’re doing here. Here’s the derivatives of each one of those pi J and K and the magnitude how you come up with that okay and so we’re going to do the derivative of this that equals this is what we found before. I drew the magnitude right here of course this is the numerator and this is the denominator.

We’re trying to and here’s the answer now you can see the answer that they have two times cosine of t minus two T sine of T okay, that’s this one here this one here over here for the J is minus two T cosine T two times sine of T okay. It’s all over four times T plus one squared now they took the square root outside the radical sign but no problem go to my site buy my program there only forty bucks and you’ll be able to do all this stuff forever you’ll notice when I. This is the greatest notebook ever these in the calculator here because my brain tells me why would I ever study something hard and not write it down somewhere where he could refer to it in the future. Well, I programmed it so in the calculator so that I can do actually do the problem no matter what variables they get. So, think about buying my programs and you’ll pass calculus you’ll get six or seven problems right in any test compared to the guy sitting next to you or the girl sitting next to you and because it’s all scored on partial credit in the class curve that’s all you need to pass the class and I never wanted to get a I did get A’s and in calculus but I never wanted to I’ve got a D I don’t care. Just get me out of there, I don’t ever let me touch this stuff again okay. Hey have a good one

Arc Length of a curve x=y^3/6+1/(2y)

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Transcripts
Hello everyone, Tom from everystepcalculus.com and everystepphysics.com. We’re talking about arc length today.  Got this on a new test that somebody sent me so I’m going to do a video of it right now. Pretty complicated nonsense in calculus but still there for calculus two, people. So let’s do it,  index 8, type in into the home screen there to get to my main menu. Your going to choose number 5, arc length and because the problem gives it an x, or a g a y.  I’m going to choose number 2. And we’re going to, I show you the formula here, here’s the formula for the arc length and the x function given is, you have to press alpha before
 you enter anything into my entry lines here.   Alpha y cubed divided by 6 plus one divided by 2 times y.  Looks good.  I always show you which of entered, you can change it if you want. I say it’s good.  I choose number 1, okay.  I always show you the derivative first; take the derivative of that. I give you the choice of which system they give you too. If they give you y, then we’re cool in this system here. We just enter what they give you. If they entered x then you have to compute the y values that were given the X values.  So we’re gonna do alpha 2 or, lower one is alpha 2, upper limit is alpha 3, or range, upper range. And I show you that also in case you made a mistake. I say  it’s ok.  We do our computations. This shows you the exact problem which is what you’re dealing with and what you write down in your paper, here.  And we’re going to square that as part of the functions, here.  You notice that calculus is so pathetic that they do one derivative and the rest of it is you have to do square roots, you have to do squares,  you have to do integrals. Well, I do that all for you. Here’s squared. Now we’re going to add one to what we’ve squared which equals this, here.   We’re going to take the square root of that whole mess right here.  And we’re going to take the integral of that which is this right here. And over this range 3 and 2. Y equals 2, substitute that in for the original equation here and you get thirteen twelves, add 3 substitute that in, you get 13 over 3, you take the upper minus the lower and you come up with thirteen over 4 or 3.25 units.  But pretty neat, huh?  everystepcalculus.com.  Go to my site and hope that I make other videos for you.  Have a good one.

Length Of Arc on the TI-89 App | Every Step Calculus Video

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Raw Transcript

This is one of my fabulous videos on length of an arc and let’s get started uhm again second, second alpha to get the letters so you can go index and then alpha eight closed parenthesis press enter and here’s my menu you can scroll up to anything you want burt we’re going to do length of an arc right now so were, that’s the number s so you can push alpha s on there too if you knew the program and needed to get there quick which I’m going to do right now. These are position vectors because they’re radius and time and ahh that’s what arc’s are and ahh this is the in vector form the r t form
r t coordinates for vectors x to the t, y to the t, z to the t and we’ll enter them now you have to push alpha again to enter the thing I got the examples up here in a test or on a homework whatever uhm let’s go alpha eight times t I always like to put times in there not just eight t calculator won’t compute that sometimes but I’ve found sometimes it didn’t so I do it all the time and then I’m safe and uhm alpha minus three times t and alpha seven times t shows you what you’ve entered here’s the vector you’re entering for the r t formula and I say it’s ok and so now we go to the r t formula which you have other ones acceleration, linear equation, parametric equations speed, unit tangent vector, velocity all these different choices right now we want length of arc so I’m going to press number three here on the calculator you can see it’s a integral with r prime t that’s the derivative, first derivative of t r t over the area of a, or the range of a and b dt and here’s the formula Length is going to be equal to this here’s the formula for it and the position vector is r t and the i coordinate the j coordinate the k coordinate and we start eight i minus three j and seventeen k keep going we have eight t and the derivative of eight t is eight derivative of minus three t is minus three and derivative of seven t is seven so you have eight squared minus three squared seven squared dt and over the time they want the range so we’re going to go alpha maybe two to the range of alpha nine shows the range over two over nine here’s the formula again put this all on your paper exactly as I show ahh the derivative of eight t is eight and the, ahh that squared is sixty four and here’s nine here’s forty nine there four sixty four, nine, forty nine and that computes out to the square root of one twenty two and one twenty two is square root of course is to the exponent one half and over nine and two area, area so substitute two for t two times this and nine times that and we have that we come up with approximately seventy seven point three the exact is seven time the square roof of one twenty two units two minus the nine that’s how you get that you always take the maximum range and subtract the minimum range from it and here’s seventy seven three eight pretty neat huh go back and do anything you want here velocity, what ever there will be another videos on those ahh everystepcalculus.com. Enjoy my programs you’ll love them.

Arc Length Calculus Video | y=sin(x) | Simpson’s Rule on the TI89

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Raw Transcript

This is a video from everystepcalculus.com demonstrating how my programs work on a TI-89, Titanium calculator and other calculators in the TI system for physics and calculus problems so we’re going to do the arc length with regards to sine or cosine and ahh first of all I’m gonna do on the calculator and show you on the calculator how this works you can do arc length here you’ll notice that when you press f three function you come up with that menu you press eight and it’ll go to arc length we’re going to put in sin of x for our function and then a comma with regard to x and then a comma with regard to the ah range of zero and pi you notice it’s busy here that means it’s thinking and the takes a while hard for the calculator to do evidently and here’s the answer three point eight two units so now let’s uhm get started on my programs your going to have to put second alpha to put the letters i n d e x in here. Then press alpha again to put nine and the open and closed parenthesispress enter and we’re into my menu we’re gonna do six which is arc length of a function of f of x which which is this which sine is and we’re gonna enter the function gonna press alpha you have to push alpha before you do anything in the entry lines here enter anything in there so alplha we’re gonna put sine of x I always ask you if it’s ok ahh case you made a mistake and so here’s the arc length formula you’ll notice you’re doing the derivative squared here and the regular function is the original function is sin of x we’re gonna enter our range here over the range of alpha zero and alpha pi I again ask you if it’s ok and then we do the derivative you write this stuff all on your paper as you see itand here’s the derivative of sine of x squared over that range here’s the simpson’s rule which you have to write on your paper I don’t think this problem would ever be on a test it’s too difficult might bemight be in your homework here’s the reality of this of that formula this took me two days to figure this out probably took simpson two years without calcuators and we’re gonna do six intervals let’s take six intervals of that change of x is equal to b minus a divided by the number of itinerations that equals point five two four and that divided by three which is simpson’s rule is point one seventy five this is the change of x and you do each one of these individually it starts with zero etcetera you keep adding this in your paper and the answer is three point eight units as you sum up all these. Pretty neat huh? everystepcalculus.com. Go to my site buy my progams and pass calculus. 

Arc Length Tutorial on the TI-89

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RawTranscript

This is a program on arc length, and let’s get started. To get my code in here to get my menu to come up, you have to put the letters i n d e x, which you do by pressing 2nd alpha, on the Titanium first for the letters and then alpha and you can enter the eight and closed parenthesis on there, press the enter button and you are into my menu. Here’s all the choices goes way down here, you can find all kinds of things on here which will help you out in your test, your homework, or just leaning about it, because its so perfect step by step, and it’s all what we wished we could have when we were looking on how to do a problem. That’s why I did it for myself first and now am offering it to you. Let’s do arc length, were going to press the number four here and get into the parametric form which is the r t form with time and radius, and three variables x y and z in this one. And this one is the Cartesian system which is equal to y = f of x, either one will give you the function there. And so we’re going to press two and enter a function. Press alpha and let’s put the function in here, we’re going to go, x cubed divided by six plus one divided by parenthesis two times x, closed parenthesis and over the range of – we have to add alpha again to get the range in here – we’re going to do it, one half to the upper range of two. It shows you what you’ve added here so you can correct it if it’s wrong or check it out here. I say it’s ok so we’re going to press one – derivative of the function here, with respect to x is x squared over two, minus one over two x squared. Here’s the formula for the arc length, the integral over the range of a and b, the square root of one plus dy dx squared, and with the respect of x. Here we have the function into the formula here, this is the part that was squared, here’s what we did the dy dx part, which we found before, mark all of this on your paper of course. Here we’ve taken the square of that, still have the square root to do, one plus, but we’re going to take the square first, like you would in normally doing a problem, here we added the one to it, here we’re going to do the square root of it, which is this. Working the formula through, here’s the, as we integrate it, here’s what the integration is over the range of two and minus one half. We put the two in the problem, two to the four minus three over six times two, which I show you here, equals thirteen twelve’s and we put the one half in here, we subtract the lower from the upper, and we added the half into the formula, here’s minus forty-seven forty-eight. And we can see that we had a minus forty seven, this will trip you up a lot of times, where there’s a minus to a minus, I do it to make sure it doesn’t happen to me, when I’m doing the problem. Thirteen twelve’s plus forty seven forty eight, the answer is thirteen sixteenths, approximately two point zero six two five, rounded to the fifth place. So pretty neat huh, everystepcalculus dot com, check out my other fabulous programs, you’ll love them, worth every penny that you buy it for, so cheap compared to the thousands of hours I’ve spent on this stuff studying it for you, to make sure they are correct and everything, and remember it encompasses calculus two and three and one, so you are buying all three semesters of your calculus in one purchase, ahh have a good one.

Arc Length

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Arc Length in Calculus with the TI-89: Raw Transcript

This is a program on arc length, and let’s get started. To get my code in here to get
my menu to come up, you have to put the letters i n d e x, which you do by pressing 2nd alpha, on the Titanium first for the letters and then alpha and you can enter the eight and
closed parenthesis on there, press the enter button and you are into my menu. Here’s
all the choices goes way down here, you can find all kinds of things on here which will
help you out in your test, your homework, or just leaning about it, because its so perfect
step by step, and it’s all what we wished we could have when we were looking on how
to do a problem. That’s why I did it for myself first and now am offering it to you.
Let’s do arc length, were going to press the number four here and get into the parametric
form which is the r t form with time and radius, and three variables x y and z in this one.
And this one is the Cartesian system which is equal to y = f of x, either one will give
you the function there. And so we’re going to press two and enter a function. Press alpha
and let’s put the function in here, we’re going to go, x cubed divided by six plus one
divided by parenthesis two times x, closed parenthesis and over the range of – we have
to add alpha again to get the range in here – we’re going to do it, one half to the
upper range of two. It shows you what you’ve added here so you can correct it if it’s
wrong or check it out here. I say it’s ok so we’re going to press one – derivative
of the function here, with respect to x is x squared over two, minus one over two x squared.
Here’s the formula for the arc length, the integral over the range of a and b, the square
root of one plus dy dx squared, and with the respect of x. Here we have the function into
the formula here, this is the part that was squared, here’s what we did the dy dx part,
which we found before, mark all of this on your paper of course. Here we’ve taken the
square of that, still have the square root to do, one plus, but we’re going to take
the square first, like you would in normally doing a problem, here we added the one to
it, here we’re going to do the square root of it, which is this. Working the formula
through, here’s the, as we integrate it, here’s what the integration is over the
range of two and minus one half. We put the two in the problem, two to the four minus
three over six times two, which I show you here, equals thirteen twelve’s and we put
the one half in here, we subtract the lower from the upper, and we added the half into
the formula, here’s minus forty-seven forty-eight. And we can see that we had a minus forty seven, this will trip you up a lot of times, where there’s a minus to a minus, I do it to make
sure it doesn’t happen to me, when I’m doing the problem. Thirteen twelve’s plus
forty seven forty eight, the answer is thirteen sixteenths, approximately two point zero six
two five, rounded to the fifth place. So pretty neat huh, everystepcalculus dot com, check
out my other fabulous programs, you’ll love them, worth every penny that you buy it for,
so cheap compared to the thousands of hours I’ve spent on this stuff studying it for
you, to make sure they are correct and everything, and remember it encompasses calculus two and three and one, so you are buying all three semesters of your calculus in one purchase, ahh have a good one.