 In this video, we'll build intuition for how we calculate arc lengths of parametrically defined curves. We'll also work through an example to calculate the arc length of such a curve. Consider the path of a plane on an air traffic controller screen that can be modeled by the parametric equations x of t being cosine of 3t and y of t being sine of 3t, where t is in minutes since the air traffic controller started observing. Suppose the air traffic controller wanted to know how far the plane has traveled in the first five minutes they begin observing. How could we use calculus to do this? We'll come back to this question in a few minutes. Let's first build some intuition into how we would calculate the length of an arc of a curve defined parametrically. Consider a curve that looks like this, for example. And I'm going to look at an interval from A to B. We're going to employ a strategy we've actually used before. Instead of trying to approximate the length of this curve by simply connecting those endpoints and finding the length of that segment, I could get a better, more accurate read of this length if I split the path into pieces. I'm going to find the length of little line segments connecting evenly spaced points. We'll call this delta x and this next point would then be A plus 2 delta x and so on. And I'm going to find the lengths of each of these paths and add them up. So let's take, for example, this segment. If I were to extend this into a right triangle, I can view this length as the hypotenuse of that right triangle. If so, I'm going to need some information about the legs of this right triangle. Well, I've made a decision to space each of these points evenly, such that this base is delta x. If that's the case, then the height of this triangle is delta y. In that case, we can actually use the Pythagorean theorem to find the hypotenuse. The hypotenuse would then be the square root of delta x squared plus delta y squared. Now, this was just some arbitrary piece of this total length. We can call this the ith length. I want to add all of those lengths up. If I do that and actually let delta x get smaller and smaller, that means I could better approximate the total length of this path if I add all of those lengths up. Ultimately, I want to find the sum of these lengths. Ultimately, since both x and y are functions of t, I'm actually going to use a little math trick. You know it as the multiplicative identity property. I'm going to bring in a multiple of 1, but I'm going to call it delta t over delta t. There's a reason for this. Since x and y are functions of t, I ultimately want to view this with respect to t. I don't want to change the value of this, so I'm going to multiply by 1, but call it delta t over delta t. With this, I'm going to bring in the denominator delta t into this square root. In doing that, this is represented by delta t squared. I just brought that delta t into the square root and to compensate, I square it, leaving me with delta t out here. I'll get a better approximation of the length of this path if I take more frequent reads of where I am and find the lengths between those points. What does it mean to take more frequent reads? That means that we're looking at the delta t, the increments of t getting smaller and smaller toward zero, and adding each of the little hypotenuses of the right triangles created by the endpoints of those reads. This is what I have. I'm going to rewrite that on the next slide using algebra. We can rewrite this. What do we recognize here? What I have is the limit of a sum. In particular, delta t is approaching zero and delta t appears in this sum as well. We know this as an integral. The first t value to the last of the square root of dx dt squared plus dy dt squared dt. Let's return now to our original question. Consider the path of a plane on an air traffic controller screen that can be modeled by the parametric equations x of t being cosine of 3t, y of t being sine of 3t. How far has the plane traveled? In the first five minutes, the controller began observing. We find our arc length, or length of path. To be the integral, now t being zero is when we initially begin observing, and we're interested in the first five minutes. We integrate from zero to five of the square root of dx dt squared plus dy dt squared. Now x was cosine of 3t and y was sine of 3t. We integrate from zero to five the square root of negative 3 sine of 3t squared plus 3 cosine of 3t squared. This is the integral from zero to five of the square root of 9 sine squared 3t plus 9 cosine squared 3t, which is the integral from zero to five of a nice, clean square root of 9. Recalling what our Pythagorean identity is, sine squared of theta plus cosine squared of theta is 1. So this is the integral from zero to five of 3dt. So the integral is 15. Now if every unit on the screen represents, let's say, 3 miles, then the plane flew a distance of 3 times 15, which is 45 miles.