- [Instructor] 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 T.
If this notion is completely unfamiliar to you,
I encourage you to review the videos on parametric equations
on Khan Academy.
But what we're going to think about
and I'm gonna talk about in generalities in this video.
In future videos we're going to be dealing
with more concrete examples
but we're gonna 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 gonna 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 would 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 wanna 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
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 notion
and I'm conceptually using the idea
of a differential as an infinitesimally small change
in that variable.
And so, this isn't 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 your rate of change
of Y with respect to T
times your change in T,
your 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 gonna be the square root
of this squared plus this squared.
So, it is the square root of,
I'm gonna give myself a little bit more space here
because I think I'm gonna 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
and then write my big radical sign,
so I'm gonna factor out a DT squared here,
so we could write this as DT squared
times X prime of T squared
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 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 all 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 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
all of these infinitesimally small changes
and that's what the definite integral does for us.
So, what we can do if we wanna add up that
plus that plus that plus that
and remember, these are infinitesimally small changes.
I'm just showing them as not infinitesimally
just so that you can kind of think about them
but if you were to add them all up,
then we are 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
feel good conceptually
for the formula of arc length
when we're dealing with parametric equations.
In the next few videos
we'll actually apply it to figure out arc lengths.