[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.57,0:00:02.78,Default,,0000,0000,0000,,- [Instructor] Let's say we're\Ngoing to trace out a curve
Dialogue: 0,0:00:02.78,0:00:05.16,Default,,0000,0000,0000,,where our X coordinate\Nand our Y coordinate
Dialogue: 0,0:00:05.16,0:00:07.33,Default,,0000,0000,0000,,that they are each defined by
Dialogue: 0,0:00:07.33,0:00:10.47,Default,,0000,0000,0000,,or they're functions\Nof a third parameter T.
Dialogue: 0,0:00:10.47,0:00:13.48,Default,,0000,0000,0000,,So, we could say that X is a function of T
Dialogue: 0,0:00:13.48,0:00:17.00,Default,,0000,0000,0000,,and we could also say\Nthat Y is a function T.
Dialogue: 0,0:00:17.00,0:00:19.56,Default,,0000,0000,0000,,If this notion is completely\Nunfamiliar to you,
Dialogue: 0,0:00:19.56,0:00:22.52,Default,,0000,0000,0000,,I encourage you to review the\Nvideos on parametric equations
Dialogue: 0,0:00:22.52,0:00:24.20,Default,,0000,0000,0000,,on Khan Academy.
Dialogue: 0,0:00:24.20,0:00:25.68,Default,,0000,0000,0000,,But what we're going to think about
Dialogue: 0,0:00:25.68,0:00:27.76,Default,,0000,0000,0000,,and I'm gonna talk about in\Ngeneralities in this video.
Dialogue: 0,0:00:27.76,0:00:29.79,Default,,0000,0000,0000,,In future videos we're going to be dealing
Dialogue: 0,0:00:29.79,0:00:32.77,Default,,0000,0000,0000,,with more concrete examples
Dialogue: 0,0:00:32.77,0:00:34.73,Default,,0000,0000,0000,,but we're gonna think\Nabout what is the path
Dialogue: 0,0:00:34.73,0:00:35.86,Default,,0000,0000,0000,,that is traced out
Dialogue: 0,0:00:35.86,0:00:38.02,Default,,0000,0000,0000,,from when T is equal to A,
Dialogue: 0,0:00:39.15,0:00:42.19,Default,,0000,0000,0000,,so this is where we are\Nwhen T is equal to A,
Dialogue: 0,0:00:42.19,0:00:45.92,Default,,0000,0000,0000,,so in this case this\Npoint would be X of A,
Dialogue: 0,0:00:45.92,0:00:47.00,Default,,0000,0000,0000,,comma Y of A,
Dialogue: 0,0:00:49.08,0:00:50.06,Default,,0000,0000,0000,,that's this point
Dialogue: 0,0:00:50.06,0:00:52.60,Default,,0000,0000,0000,,and then as we increase from T equals A
Dialogue: 0,0:00:52.60,0:00:54.19,Default,,0000,0000,0000,,to T is equal to B,
Dialogue: 0,0:00:55.82,0:00:59.67,Default,,0000,0000,0000,,so our curve might do something like this,
Dialogue: 0,0:00:59.67,0:01:02.36,Default,,0000,0000,0000,,so this is when T is equal to B,
Dialogue: 0,0:01:02.36,0:01:03.22,Default,,0000,0000,0000,,T is equal to B,
Dialogue: 0,0:01:03.22,0:01:06.09,Default,,0000,0000,0000,,so this point right over here is X of B,
Dialogue: 0,0:01:06.09,0:01:07.17,Default,,0000,0000,0000,,comma Y of B.
Dialogue: 0,0:01:08.73,0:01:10.61,Default,,0000,0000,0000,,Let's think about how do we figure out
Dialogue: 0,0:01:10.61,0:01:12.81,Default,,0000,0000,0000,,the length of this actual curve,
Dialogue: 0,0:01:12.81,0:01:17.31,Default,,0000,0000,0000,,this actual arc length from\NT equals A to T equals B?
Dialogue: 0,0:01:17.31,0:01:18.40,Default,,0000,0000,0000,,Well, to think about that
Dialogue: 0,0:01:18.40,0:01:20.69,Default,,0000,0000,0000,,we're gonna zoom in and\Nthink about what happens
Dialogue: 0,0:01:20.69,0:01:24.55,Default,,0000,0000,0000,,when we have a very small change in T?
Dialogue: 0,0:01:24.55,0:01:26.04,Default,,0000,0000,0000,,So, a very small change in T.
Dialogue: 0,0:01:26.04,0:01:28.62,Default,,0000,0000,0000,,Let's say we're starting at\Nthis point right over here
Dialogue: 0,0:01:28.62,0:01:30.64,Default,,0000,0000,0000,,and we have a very small change in T,
Dialogue: 0,0:01:30.64,0:01:34.96,Default,,0000,0000,0000,,so we go from this point\Nto let's say this point
Dialogue: 0,0:01:34.96,0:01:36.80,Default,,0000,0000,0000,,over that very small change in T.
Dialogue: 0,0:01:36.80,0:01:38.09,Default,,0000,0000,0000,,It actually would be\Nmuch smaller than this
Dialogue: 0,0:01:38.09,0:01:39.10,Default,,0000,0000,0000,,but if I drew it any smaller,
Dialogue: 0,0:01:39.10,0:01:40.49,Default,,0000,0000,0000,,you would have trouble seeing it.
Dialogue: 0,0:01:40.49,0:01:43.08,Default,,0000,0000,0000,,So, let's say that that\Nis our very small change
Dialogue: 0,0:01:43.08,0:01:46.75,Default,,0000,0000,0000,,in our path in our arc\Nthat we are traveling
Dialogue: 0,0:01:47.87,0:01:50.18,Default,,0000,0000,0000,,and so, we wanna find this length.
Dialogue: 0,0:01:50.18,0:01:52.25,Default,,0000,0000,0000,,Well, we could break it down
Dialogue: 0,0:01:52.25,0:01:54.24,Default,,0000,0000,0000,,into how far we've\Nmoved in the X direction
Dialogue: 0,0:01:54.24,0:01:56.60,Default,,0000,0000,0000,,and how far we've moved\Nin the Y direction.
Dialogue: 0,0:01:56.60,0:01:58.79,Default,,0000,0000,0000,,So, in the X direction,
Dialogue: 0,0:01:58.79,0:02:00.49,Default,,0000,0000,0000,,the X direction right over here,
Dialogue: 0,0:02:00.49,0:02:03.00,Default,,0000,0000,0000,,we would have moved a\Nvery small change in X
Dialogue: 0,0:02:03.00,0:02:04.64,Default,,0000,0000,0000,,and what would that be equal to?
Dialogue: 0,0:02:04.64,0:02:05.82,Default,,0000,0000,0000,,Well, that would be the rate of change
Dialogue: 0,0:02:05.82,0:02:08.79,Default,,0000,0000,0000,,with which we are\Nchanging with respect to T
Dialogue: 0,0:02:08.79,0:02:11.96,Default,,0000,0000,0000,,with which X is changing with respect to T
Dialogue: 0,0:02:11.96,0:02:14.37,Default,,0000,0000,0000,,times our very small change in T
Dialogue: 0,0:02:14.37,0:02:15.86,Default,,0000,0000,0000,,and this is a little hand wavy,
Dialogue: 0,0:02:15.86,0:02:17.67,Default,,0000,0000,0000,,I'm using differential notion
Dialogue: 0,0:02:17.67,0:02:20.00,Default,,0000,0000,0000,,and I'm conceptually using the idea
Dialogue: 0,0:02:20.00,0:02:23.81,Default,,0000,0000,0000,,of a differential as an\Ninfinitesimally small change
Dialogue: 0,0:02:23.81,0:02:25.73,Default,,0000,0000,0000,,in that variable.
Dialogue: 0,0:02:25.73,0:02:27.45,Default,,0000,0000,0000,,And so, this isn't a formal proof
Dialogue: 0,0:02:27.45,0:02:29.01,Default,,0000,0000,0000,,but it's to give us the intuition
Dialogue: 0,0:02:29.01,0:02:30.90,Default,,0000,0000,0000,,for how we derive arc length
Dialogue: 0,0:02:30.90,0:02:33.16,Default,,0000,0000,0000,,when we're dealing with\Nparametric equations.
Dialogue: 0,0:02:33.16,0:02:35.74,Default,,0000,0000,0000,,So, this will hopefully\Nmake conceptual sense
Dialogue: 0,0:02:35.74,0:02:36.87,Default,,0000,0000,0000,,that this is our DX.
Dialogue: 0,0:02:36.87,0:02:38.18,Default,,0000,0000,0000,,In fact, we could even write it this way,
Dialogue: 0,0:02:38.18,0:02:42.35,Default,,0000,0000,0000,,DX/DT, that's the same thing\Nas X prime of T times DT
Dialogue: 0,0:02:43.43,0:02:44.75,Default,,0000,0000,0000,,and then our change in Y
Dialogue: 0,0:02:44.75,0:02:47.16,Default,,0000,0000,0000,,is going to be the same idea.
Dialogue: 0,0:02:47.16,0:02:49.69,Default,,0000,0000,0000,,Our change in Y, our\Ninfinitesimally small change in Y
Dialogue: 0,0:02:49.69,0:02:52.12,Default,,0000,0000,0000,,when we have an infinitesimally\Nsmall change in T,
Dialogue: 0,0:02:52.12,0:02:53.48,Default,,0000,0000,0000,,well, you could view that\Nas your rate of change
Dialogue: 0,0:02:53.48,0:02:55.39,Default,,0000,0000,0000,,of Y with respect to T
Dialogue: 0,0:02:55.39,0:02:57.19,Default,,0000,0000,0000,,times your change in T,
Dialogue: 0,0:02:57.19,0:02:59.28,Default,,0000,0000,0000,,your very small change in T
Dialogue: 0,0:02:59.28,0:03:00.59,Default,,0000,0000,0000,,which is going to be equal to,
Dialogue: 0,0:03:00.59,0:03:03.84,Default,,0000,0000,0000,,we could write that as Y prime of T DT.
Dialogue: 0,0:03:05.35,0:03:07.82,Default,,0000,0000,0000,,Now, based on this,\Nwhat would be the length
Dialogue: 0,0:03:07.82,0:03:12.50,Default,,0000,0000,0000,,of our infinitesimally small\Narc length right over here?
Dialogue: 0,0:03:12.50,0:03:15.22,Default,,0000,0000,0000,,Well, that we could just\Nuse the Pythagorean theorem.
Dialogue: 0,0:03:15.22,0:03:18.39,Default,,0000,0000,0000,,That is going to be the square root of,
Dialogue: 0,0:03:18.39,0:03:20.09,Default,,0000,0000,0000,,that's the hypotenuse\Nof this right triangle
Dialogue: 0,0:03:20.09,0:03:21.00,Default,,0000,0000,0000,,right over here.
Dialogue: 0,0:03:21.00,0:03:22.01,Default,,0000,0000,0000,,So, it's gonna be the square root
Dialogue: 0,0:03:22.01,0:03:24.36,Default,,0000,0000,0000,,of this squared plus this squared.
Dialogue: 0,0:03:24.36,0:03:26.50,Default,,0000,0000,0000,,So, it is the square root of,
Dialogue: 0,0:03:26.50,0:03:28.39,Default,,0000,0000,0000,,I'm gonna give myself a\Nlittle bit more space here
Dialogue: 0,0:03:28.39,0:03:30.11,Default,,0000,0000,0000,,because I think I'm gonna use a lot of it,
Dialogue: 0,0:03:30.11,0:03:32.19,Default,,0000,0000,0000,,so the stuff in blue squared,
Dialogue: 0,0:03:32.19,0:03:36.22,Default,,0000,0000,0000,,DX squared we could\Nrewrite that as X prime
Dialogue: 0,0:03:36.22,0:03:37.47,Default,,0000,0000,0000,,of T DT squared
Dialogue: 0,0:03:39.37,0:03:43.54,Default,,0000,0000,0000,,plus this squared which\Nis Y prime of T DT squared
Dialogue: 0,0:03:47.98,0:03:50.52,Default,,0000,0000,0000,,and now let's just try to\Nsimplify this a little bit.
Dialogue: 0,0:03:50.52,0:03:52.96,Default,,0000,0000,0000,,Remember, this is this\Ninfinitesimally small arc length
Dialogue: 0,0:03:52.96,0:03:54.44,Default,,0000,0000,0000,,right over here.
Dialogue: 0,0:03:54.44,0:03:58.80,Default,,0000,0000,0000,,So, we can actually\Nfactor out a DT squared,
Dialogue: 0,0:03:58.80,0:04:00.65,Default,,0000,0000,0000,,it's a term in both of these
Dialogue: 0,0:04:00.65,0:04:02.98,Default,,0000,0000,0000,,and so, we can rewrite this as,
Dialogue: 0,0:04:02.98,0:04:05.54,Default,,0000,0000,0000,,let me, so I can rewrite this
Dialogue: 0,0:04:05.54,0:04:07.64,Default,,0000,0000,0000,,and then write my big radical sign,
Dialogue: 0,0:04:07.64,0:04:10.83,Default,,0000,0000,0000,,so I'm gonna factor out a DT squared here,
Dialogue: 0,0:04:10.83,0:04:14.22,Default,,0000,0000,0000,,so we could write this as DT squared
Dialogue: 0,0:04:14.22,0:04:16.39,Default,,0000,0000,0000,,times X prime of T squared
Dialogue: 0,0:04:19.83,0:04:21.91,Default,,0000,0000,0000,,plus Y prime of T squared
Dialogue: 0,0:04:26.45,0:04:27.28,Default,,0000,0000,0000,,and then to be clear
Dialogue: 0,0:04:27.28,0:04:29.85,Default,,0000,0000,0000,,this is being multiplied\Nby all of this stuff
Dialogue: 0,0:04:29.85,0:04:31.07,Default,,0000,0000,0000,,right over there.
Dialogue: 0,0:04:31.07,0:04:34.62,Default,,0000,0000,0000,,Well, now if we have this DT\Nsquared under the radical,
Dialogue: 0,0:04:34.62,0:04:35.60,Default,,0000,0000,0000,,we can take it out
Dialogue: 0,0:04:35.60,0:04:36.91,Default,,0000,0000,0000,,and so, we will have a DT
Dialogue: 0,0:04:36.91,0:04:40.75,Default,,0000,0000,0000,,and so, this is all going to\Nbe equal to the square root
Dialogue: 0,0:04:40.75,0:04:44.13,Default,,0000,0000,0000,,of, so the stuff that's\Nstill under the radical
Dialogue: 0,0:04:44.13,0:04:47.04,Default,,0000,0000,0000,,is going to be X prime of T squared
Dialogue: 0,0:04:50.38,0:04:52.46,Default,,0000,0000,0000,,plus Y prime of T squared
Dialogue: 0,0:04:55.41,0:04:57.52,Default,,0000,0000,0000,,and now we took out a DT
Dialogue: 0,0:04:57.52,0:04:59.86,Default,,0000,0000,0000,,and now we took out a DT.
Dialogue: 0,0:04:59.86,0:05:03.03,Default,,0000,0000,0000,,I could have written it right over here
Dialogue: 0,0:05:03.03,0:05:04.26,Default,,0000,0000,0000,,but I'm just writing it on the other side,
Dialogue: 0,0:05:04.26,0:05:06.27,Default,,0000,0000,0000,,we're just multiplying the two.
Dialogue: 0,0:05:06.27,0:05:08.66,Default,,0000,0000,0000,,So, this is once again just\Nrewriting the expression
Dialogue: 0,0:05:08.66,0:05:12.83,Default,,0000,0000,0000,,for this infinitesimally\Nsmall change in arc length.
Dialogue: 0,0:05:12.83,0:05:15.51,Default,,0000,0000,0000,,Well, what's lucky for us is in calculus
Dialogue: 0,0:05:15.51,0:05:17.86,Default,,0000,0000,0000,,we have the tools for adding up
Dialogue: 0,0:05:17.86,0:05:21.56,Default,,0000,0000,0000,,all of these infinitesimally small changes
Dialogue: 0,0:05:21.56,0:05:24.14,Default,,0000,0000,0000,,and that's what the definite\Nintegral does for us.
Dialogue: 0,0:05:24.14,0:05:26.18,Default,,0000,0000,0000,,So, what we can do if we wanna add up that
Dialogue: 0,0:05:26.18,0:05:27.93,Default,,0000,0000,0000,,plus that plus that plus that
Dialogue: 0,0:05:27.93,0:05:30.02,Default,,0000,0000,0000,,and remember, these are\Ninfinitesimally small changes.
Dialogue: 0,0:05:30.02,0:05:32.32,Default,,0000,0000,0000,,I'm just showing them\Nas not infinitesimally
Dialogue: 0,0:05:32.32,0:05:34.28,Default,,0000,0000,0000,,just so that you can\Nkind of think about them
Dialogue: 0,0:05:34.28,0:05:35.86,Default,,0000,0000,0000,,but if you were to add them all up,
Dialogue: 0,0:05:35.86,0:05:38.61,Default,,0000,0000,0000,,then we are essentially\Ntaking the integral
Dialogue: 0,0:05:38.61,0:05:40.65,Default,,0000,0000,0000,,and we're integrating with respect to T
Dialogue: 0,0:05:40.65,0:05:43.86,Default,,0000,0000,0000,,and so, we're starting at T is equal to A,
Dialogue: 0,0:05:43.86,0:05:46.60,Default,,0000,0000,0000,,all the way to T is equal to B
Dialogue: 0,0:05:46.60,0:05:50.60,Default,,0000,0000,0000,,and just like that we\Nhave been able to at least
Dialogue: 0,0:05:51.47,0:05:52.93,Default,,0000,0000,0000,,feel good conceptually
Dialogue: 0,0:05:52.93,0:05:55.52,Default,,0000,0000,0000,,for the formula of arc length
Dialogue: 0,0:05:55.52,0:05:59.46,Default,,0000,0000,0000,,when we're dealing with\Nparametric equations.
Dialogue: 0,0:05:59.46,0:06:00.84,Default,,0000,0000,0000,,In the next few videos
Dialogue: 0,0:06:00.84,0:06:04.22,Default,,0000,0000,0000,,we'll actually apply it\Nto figure out arc lengths.