§8.1 & §10.2: Arc Length
Resources for computing arc length in the three main forms used in class: $y=f(x)$, $x=g(y)$, and parametric curves given by $x=f(t)$ and $y=g(t)$. The emphasis here is on choosing the correct formula, setting up the integral cleanly, and handling algebra carefully. This page intentionally stays focused on arc length only, not tangent lines, speed, curvature, or surface area.
Library, Handouts & Tools
| Resource Name | Type | Description | Link |
|---|---|---|---|
| OpenStax: Arc Length of a Curve | Textbook Notes | Clear derivation and examples for arc length when the curve is written as $y=f(x)$ and when it is written as $x=g(y)$. | Read Notes |
| OpenStax: Calculus of Parametric Curves | Textbook Notes | Includes the parametric arc length formula and explains how it connects back to the usual arc length formula from single-variable calculus. | Read Notes |
| Paul's Notes: Arc Length | Study Guide | Good for standard Calc II arc length setup, including both $y=f(x)$ and $x=g(y)$, with worked examples that are very classroom-friendly. | Read Notes |
| Paul's Notes: Arc Length with Parametric Equations | Study Guide | Focused specifically on arc length for curves given by $x=f(t)$ and $y=g(t)$, including the important warning about tracing a curve more than once. | Read Notes |
| OpenStax: Key Equations | Formula Sheet | Quick formula reference for arc length when the curve is written as a function of $x$ or as a function of $y$. | Open Sheet |
Video Lectures
| Topic | Source | Description | Link |
|---|---|---|---|
| Video Calculus Series (Arc Length + Parametric Unit) | Houston ACT | The UH Video Calculus index includes arc length for ordinary curves and also a parametric unit covering slope, arc length, and area. | Open Index |
| Arc Length Introduction (Calculus 2) | Houston Math Prep | A short introductory video on the basic arc length setup for functions written in the usual single-variable form. | Watch |
| Arc Length Calculus (Example 1) | Houston Math Prep | A worked example that is useful once you already know the formula and want to see the algebra and antiderivative handled carefully. | Watch |
| Worked Example: Arc Length | Khan Academy | A clean single-variable example for the standard formula $L=\int_a^b \sqrt{1+[f'(x)]^2}\,dx$. | Watch |
| Parametric Curve Arc Length | Khan Academy | Conceptual introduction to the formula $L=\int_\alpha^\beta \sqrt{(x'(t))^2+(y'(t))^2}\,dt$ for curves given parametrically. | Watch |
| Worked Example: Parametric Arc Length | Khan Academy | A full example showing how the parametric formula is set up and simplified on a concrete curve. | Watch |
Practice & Review
| Topic | Source | Description | Link |
|---|---|---|---|
| Arc Length Practice Problems | Paul's Notes | Extra practice on standard arc length problems, including the right formula choice and the algebra under the radical. | Practice |
| Parametric Arc Length Practice Problems | Paul's Notes | Good mixed practice for curves in parametric form, especially when you need to simplify $\sqrt{(x'(t))^2+(y'(t))^2}$. | Practice |
| Parametric Curve Arc Length Practice | Khan Academy | Quick online practice if you want a few extra parametric arc length questions with immediate feedback. | Practice |
| Formula Check: Arc Length of $x=g(y)$ | OpenStax | A quick place to revisit the less-common $L=\int_c^d \sqrt{1+[g'(y)]^2}\,dy$ formula when the curve is easier to describe in terms of $y$. | Review |