#8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled

In this section of the AP Calculus BC course, we will learn how to find the length of a smooth, planar curve, also known as the arc length. Then, we will look at how we can find the distance an object has traveled using the arc length of a curve. Finally, we’ll provide some practice problems so you can master this topic. Let’s get started!

#🎢 What is Arc Length?

Before we dive into the process of computing arc length, let's clarify the fundamental concept behind it. In calculus, the arc length of a curve is the measure of the distance along the curve. You can think of it as tracing the curve with a string and then straightening the string—the length of that string is the arc length. For example, consider a roller coaster at an amusement park. The total length of the track of the roller coaster—including all the crazy twists, turns, and loop-the-loops—is the arc length of that track. Each segment of the track contributes to the overall arc length, offering a precise way to quantify the distance covered during the exhilarating ride.

!Arc_length.gif

Visualization of arc length.

GIF courtesy of Wikipedia.

#〰️ How To Find the Arc Length of a Curve

Now, let's equip ourselves with the mathematical tools to calculate the arc length of a function. The arc length $S$ of a **smooth, planar ...