Chapter 10: Parametric Equations and Polar Coordinates

Students learn to eliminate the parameter to convert between parametric and Cartesian forms, as well as how to determine domain, range, and curve orientation. Calculus is applied to parametric equations, with derivatives used to find slopes of tangents, rates of change, and concavity, and integrals used to compute arc length and areas under parametric curves. The chapter introduces polar coordinates, where points are described by a distance 𝑟 r from the origin and an angle 𝜃θ, offering a natural framework for analyzing curves with rotational symmetry. Converting between polar and Cartesian coordinates is covered, along with graphing polar equations such as limacons, cardioids, and roses. The chapter explains how to calculate areas enclosed by polar curves and arc lengths in polar form. Applications in physics and engineering—such as analyzing planetary orbits, modeling motion in circular paths, and describing periodic phenomena—demonstrate the power of these alternative coordinate systems. By mastering parametric and polar methods, students gain flexible tools for modeling complex curves and solving real-world problems beyond the reach of standard Cartesian techniques.