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

Abigail Young

Abigail Young

7 min read

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Study Guide Overview

This study guide covers arc length of smooth, planar curves and distance traveled. It explains the concept of arc length, provides the formula: S = ∫ₐᵇ √(1 + [f'(x)]²) dx, and breaks down its components. It also demonstrates how to use arc length to calculate distance traveled with examples and practice problems involving finding the arc length and distance traveled by a particle along a curve.

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 SS of a **smooth, planar ...

Question 1 of 11

Arc length measures the 📏...

area under a curve

slope of the tangent line

distance along a curve

rate of change of a function