ยง10.4: Calculus in Polar Coordinates
Resources for the first calculus topics in polar coordinates: areas of polar regions and tangent lines to polar curves. The focus here is on setting up and evaluating area integrals, finding intersection angles, computing $\dfrac{dy}{dx}$ for curves given by $r=f(\theta)$, and identifying horizontal and vertical tangents. This page intentionally excludes polar arc length.
Library, Handouts & Tools
| Resource Name | Type | Description | Link |
|---|---|---|---|
| OpenStax Calculus: Area in Polar Coordinates | Textbook Notes | Best single reference for the core area formulas in polar form, including the area enclosed by one polar curve and the area between two polar curves. | Read Notes |
| Paul's Notes: Tangents with Polar Coordinates | Derivative Guide | Clear explanation of how to treat a polar curve as the parametric pair $x=r\cos\theta$, $y=r\sin\theta$, compute $\dfrac{dy}{dx}$, and locate horizontal and vertical tangents. | Read Notes |
| Paul's Notes: Area with Polar Coordinates | Area Guide | Very useful for setting bounds, recognizing the geometry of a polar region, and handling inside-versus-outside problems with one or two polar curves. | Read Notes |
| Paul's Notes: Parametric & Polar Chapter Overview | Section Hub | Helpful chapter-level index when you want the polar tangent and polar area sections collected in one place without digging around. | Open Hub |
| Desmos: Polar Graphing Help | Tool Guide | Use this when you need to turn on the polar grid, graph two polar curves together, and visually identify the correct region or intersection angles before setting up integrals. | Open Guide |
| Desmos: Polar Coordinates Demo | Interactive | A simple polar graph you can edit directly to test area regions, inspect overlap, and check where tangent behavior changes. | Open Graph |
Video Lectures
| Topic | Source | Description | Link |
|---|---|---|---|
| Video Calculus Series (Parametric & Polar Sequence) | Houston ACT | The UH Video Calculus index is a useful hub for the polar unit. For this section, use only the area and tangent-line portions and skip any polar arc-length videos. | Open Page |
| Area Bounded by Polar Curves | Khan Academy | Good conceptual video for where the $\tfrac12\int r^2\,d\theta$ formula comes from and how to think about a region traced in polar form. | Watch |
| Worked Example: Differentiating Polar Functions | Khan Academy | A targeted worked example for computing $\dfrac{dy}{dx}$ from a polar equation, which is the key step behind tangent-line questions in this section. | Watch |
| AP Calculus BC Polar Unit | Khan Academy | Unit page collecting the polar tangent and polar area lessons in one place. Useful if you want to work through the section in sequence. | Open Unit |
Focused Practice Path
| Skill | Best Resource | What to Focus On | Link |
|---|---|---|---|
| Set up the area inside one polar curve | Khan Academy + OpenStax | Practice deciding the correct $\theta$-interval and applying $A=\tfrac12\int_a^b r^2\,d\theta$ without accidentally integrating extra petals or repeated traces. | Practice |
| Find area between two polar curves | OpenStax + Paul's Notes | Focus on intersection angles first, then identify which graph is outer and which is inner on the relevant interval before subtracting the two area pieces. | Practice |
| Compute $\dfrac{dy}{dx}$ for $r=f(\theta)$ | Paul's Notes | Translate the polar curve into $x=r\cos\theta$ and $y=r\sin\theta$, differentiate each with respect to $\theta$, and form $\dfrac{dy/d\theta}{dx/d\theta}$ carefully. | Read Notes |
| Find horizontal and vertical tangents | Khan Academy | After finding $\dfrac{dy}{dx}$, separate numerator-zero cases from denominator-zero cases and check for any overlap or undefined points before concluding tangent behavior. | Practice |
| Check regions and tangent behavior visually | Desmos | Use graphing to confirm what the region actually looks like, estimate intersection values, and sanity-check whether your tangent calculations match the shape of the curve. | Open Graph |