Course: Calculus II
Topic: Parametric, Polar, and Conic Curves
Subtopic: Polar Equations of Conic Sections


Start by reviewing conic sections from a pre-calculus course. See Lesson Notes - College Algebra: Introduction to Conics | Parabolas | Ellipses | Hyperbolas | General & Degenerate Conics but ignore any "text notes". Work conics that are centered at the origin as well as those translated away from the origin.

Our main goal in this lesson is to concentrate performing calculus on the conic sections. Many calculus operations are simplified by converting the equations of the conics from rectangular form to polar form. This transformation is accomplished through an ingenious formula. We can then apply calculus to the polar form to find the area inside a bounded conic, the arc length of the conic, etc. Study conics that have horizontal or vertical axes as well as those that have a rotated axis.


By the end of this topic you should know and be prepared to be tested on:

Terminology & Formulae

Terms you should be able to define: focal point, major and minor axes, directrix, pole, eccentricity "e", Kepler and his laws

Equation of polar form of conic section. Graphic identifies various cues obtained from formula such as sine means horizontal directrix but cosine means vertical.

Mini-Lectures and Examples

STUDY: Polar Equations of Conic Sections

Supplemental Resources (recommended)

As a review of Conic Sections from Pre-Calculus, you may want to download one of the following formula sheets, whichever works best for you:

Watch The Organic Chemistry Tutor's Calculus II video Eccentricity of an Ellipse and/or explore the connection between eccentricity and conics via the interactive graph at Math Is Fun: Eccentricity.

Supplemental Resources (optional)

These are both quite good reads. The first reviews concepts from Pre-Calculus and the second covers Calculus concepts.

Lesson: Conic Sections, Dale Hoffman's Contemporary Calculus

Lesson: Properties of the Conic Sections, Dale Hoffman's Contemporary Calculus

This video was recommended by a student who found it helped pull some Pre-Calculus concepts together with the new way of looking at conics via eccentricity: Thinkwell Vids: The Eccentricity of an Ellipse.

rev. 2021-03-06