Parametric curve arc length

- [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.