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### Course: AP®︎/College Calculus BC > Unit 9

Lesson 3: Finding arc lengths of curves given by parametric equations# Parametric curve arc length

Conceptual introduction to the formula for arc length of a parametric curve.

## Want to join the conversation?

- Why doesn't the integral give us the area under the curve?(11 votes)
- Normally, you're integrating a quantity like f(x) dx, which is a rectangle of height f(x) and infinitesimal width dx, so the sum of infinitely many rectangles gives you area. However, here you're integrating a quantity ds = sqrt(dx^2 + dy^2), which is a line segment of infinitesimal length; the sum of infinitely many line segments gives you length. Let me know if this helps.(14 votes)

- Hi, About 2 minutes in to the video Sal explains that dx=(dx/dt)*dt and that dy=(dx/dt)*dt. I have a hard time understanding how that works and why that is the case. Can someone help me understand this better?(10 votes)
- In the video, Dx is the rate of change our function X. Our function X is written in terms of t, so the derivative of X(t) will be dx/dt, the derivative of our function X with respect to t, multiplied by dt, the derivative or rate of change of the variable t, which will always be equal to 1 here. It's basically the same thing as taking the derivative of any other function with the variable x in it, but in this case its replaced with the variable t. For example, the derivative of x^2 is equal to 2x(dx) , where d/dx=2x and dx=1. So in the video, dx/dt is like d/dx and dt=dx.(3 votes)

- It doesn't make sense to me conceptually that dx=(dx/dt)*dt. Why are we multiplying times dt, besides the fact that it cancels out with dx/dt to give dx? Is there another reason besides this, or is that the reason? Please explain.(6 votes)
- That is the reason. The thing is that technically you are not supposed to treat the dx and dt like quantities but we do it because it is convenient and easy. Hope this helps!(1 vote)

- Why can't you just find dy/dx (by doing (dy/dt)/(dx/dt) then use the arc length equation: int(sqrt(1+(dy/dx)^2)). You can also find your limits by plugging the starting and end point of the interval in your paramteric equations and solving.(4 votes)
- I think the reason is because we cannot integrate that equation with respect to dx because the function is in terms of t not x, we need to integrate the equation with respect to dt and that's why we did all that clever math manipulations.(1 vote)

- Why didn't we solve it by turning the parametric equation into y(x)= expression with x? Would it still work if we did like that or is it problematic because we lose the information of time related to position?(1 vote)
- This isn't always possible because y may not be a function of x. Consider the curve given by

<x, y>=<tcos(t), tsin(t)>. This is a spiral centered on the origin, so it fails both the vertical line test and the horizontal line test infinitely many times.

We use parametric equations because there are lots of curves that just can't be described by y as a function of x. This gives us a more powerful language, but it also means we can't always convert back to the 'familiar' y=f(x) setting.(6 votes)

- What is the formula of area under a parametric curve ?(2 votes)
- If x=f(t) and y=g(t) and the parametric curve is traced out exactly once as t increases from a to b, then the area is the integral from a to b of g(t)*f'(t) dt (not sure how to do integral formatting).(1 vote)

- can we use the arc length formula to get the arc length here? if no, why not?(1 vote)
- By arc length formula, do you mean sqrt(1+(dy/dx)^(2))? If so, see that to use that, you need dy/dx. For that, you need to use x(t) and y(t) to find y(x). That works too though, You're still bound to get the same answer.

However, this'll only work if you're able to express y as a function of x. There could be cases where you can't. In such cases, using this formula would be the better choice.(2 votes)

- This is a moment where I was in a classroom. My brain is on fire! There's a relationship that I can almost see, but can't quite make it out.

Integration of a function is used to calculate the accumulated change between two bounds. But that same mechanism of integration is being used to calculate the length of the arc. We're just taking square roots via the Pythagorean theorem.

There's some magic relationship now between these pythagorean square roots and the accumulated change. Yes, the function being integrated is different. But it's related. Which means there is a relationship between the pythagorean theorem and the area under the curve; the accumulated change.

I asked ChatGPT, but it's not smart enough. I need a real teacher! :)(0 votes)- The relationship that you're seeing isn't between the Pythagorean theorem and the area under the curve, per se.

First, don't think 'Pythagorean theorem', think 'distance formula'. The square root of the sum of the squares is just how distance is defined on the coordinate plane. And of course the distance formula is here, we're measuring a length.

Second, move away from the idea of integration as 'area under the curve'. Think of it more generally as the total 'amount' of a function over a region. So far, that region has usually been just an interval of the real numbers, a straight line with no curvature to account for, which is why the 'area under the curve' idea has worked so far. We visualize the 'amount' of the function in the y-direction when we graph it, and then taking that area is equivalent to summing up the function in that interval. (And look at what happens if the red curve is just a horizontal line with y'(t)=0; this square-root integral expression just reduces to x(b)-x(a), the total distance traveled.)

But now, we're summing functions over more elaborate curves, and so we use the distance formula to take the now-harder-to-measure length of the path into account. In this video, the function we're implicitly summing up is the constant f=1; we're just taking the length and not changing it.

But in short, we see the concepts of the distance formula and integration together because we explicitly put them together; we're computing arc length by adding up a bunch of small distances.

Also, you're correct that ChatGPT is not smart enough. I would advise you against taking anything it produces as truth, because it has no concept of true or false statements and will say them both with equal confidence. It is essentially a very large autocomplete.(3 votes)

- formula for arc length along a parametric curve(0 votes)

## Video transcript

- [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.