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# Parametric curve arc length

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## Video transcript

let's say we're going to trace out a curve where our x-coordinate and our y-coordinate that they are each defined by or they're functions of a third parameter T so we could say that X is a function of T and we could also say that Y is a function of T if this notion is completely unfamiliar to you I encourage you to review the videos on parametric equations on Khan Academy but we're going to think about I'm going to talk about it in generalities in this video in future videos we're going to be dealing with more concrete examples but we're going to think about what is the path that is traced out from when T is equal to a so this is where we are when T is equal to a so in this case this point would be X of a comma Y of a that's this point and then as we increase from T equals a to T is equal to B so our curve might do something like this so this is when T is equal to B T is equal to B so this point right over here is X of B comma Y of B let's think about how do we figure out the length of this actual curve this actual arc length from T equals a to T equals B well to think about that we're going to zoom in and think about what happens when we have a very small change in T so a very small change in T let's say we're starting at this point right over here and we have a very small change in T so we go from this point to let's say this point over that very small change in T it actually be much smaller than this but if I drew it any smaller you would have trouble seeing it so let's say that that is our very small change in our path in our arc that we are traveling and so we want to find this length well we could break it down into how far we've moved in the X direction and how far we've moved in the Y direction so in the X direction the X Direction right over here we would have moved a very small change in X and what would that be equal to well that would be the rate of change with which we are changing with respect to T which with which X is changing with respect to T times our very small change in T and this is a little hand wavy I'm using differential notation and I'm conceptually using the idea of a differential as an infinitesimally small change in that variable but this and so this is it's a formal proof but it's to give us the intuition for how we derive arc length when we're dealing with parametric equations so this will hopefully make conceptual sense that this is our DX in fact we could even write it this way DX DT that's the same thing as X prime of T times DT and then our change in Y is going to be the same idea our change in Y our infinitesimally small change in Y when we have an infinitesimally small change in T well you could view that as a rate of change of Y with respect to T times your change in T or very small change in T which is going to be equal to we could write that as Y prime of T DT now based on this what would be the length of our infinitesimally small arc length right over here well that we could just use the Pythagorean theorem that is going to be the square root of that's the hypotenuse of this right triangle right over here so it's going to be the square root of this squared plus this squared so it is the square root of I'm going to give myself a little bit more space here because I think I'm going to use a lot of it so the stuff in blue squared DX squared we could rewrite that as X prime of T DT squared plus this squared which is y prime of T DT squared and now let's just try to simplify this a little bit remember this is this infinitesimally small arc length right over here so we can actually factor out a DT squared it's a term in both of these and so we can rewrite this as let me so I can rewrite this let my big radical sign so I'm going to factor out a DT squared here so we could write this as DT squared times X prime of T squared plus plus y prime of T squared and then to be clear this is being multiplied by all of this stuff right over there well now we can actually if we have this DT squared under the radical we can take it out and so we will have a DT and so this is going to be equal to the square root of so the stuff that's still under the radical is going to be X prime of T squared plus y prime of T squared and now we took out a DT and now we took out a DT I could have I could have written it right over here but I'm just writing it on the other side we're just multiplying the two so this is once again just rewriting the expression for this infinitesimally small change in arc length well what's lucky for us is in calculus we have the tools for adding up and it all of these infinitesimally small changes that's what the definite integral does for us so what we can do if we want to add up that Plus that Plus that Plus that in remember these are infinitesimally small changes you I'm just showing them as not infinitesimal just so you can kind of think about them but if you were to add them all up that we're essentially taking the integral and we're integrating with respect to T and so we're starting at T is equal to a all the way to T is equal to B and just like that we have been able to at least get a conceptual get feel good conceptually for the formula of arc length when we're dealing with parametric equations in the next two videos we'll actually apply it to figure out arc lengths
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