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


Recall from a Pre-Calculus course parametric equations (see Lesson Notes - Trigonometry - Parametric Equations and Graphs but ignore any "text notes"). A parametric equation, e.g. `x=t^3`, `y=tan(t)`, can be converted to a rectangular equation (the previous example would be `y=tan(root3(x))`, but some calculus operations may be simplified by working with the parametric form. All the calculus we have done with rectangular equations we'll redo in this lesson but using parametric form. We'll find derivatives and integrals of parametric equations, slopes of tangent lines to parametric curves, arc lengths of parametric curves, and more.


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


Terms you should be able to define: parametric equation, parameter

Mini-Lectures and Examples

STUDY: Parametric Equations and Curves

Supplemental Resources (recommended)

Simple sheet with Calculus Formulas for Parametric Equations from Dr. Rahman, Univ. of N. Florida.

Supplemental Resources (optional)

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

Lesson: Parametric Equations, Dale Hoffman's Contemporary Calculus

Lesson: Calculus of Parametric Equations, Dale Hoffman's Contemporary Calculus

This article about Ocean Waves and the Trochoid Curve provides a nice little application of a particular parametric curve.

Bezier curves are not covered in Calculus II, but they are a beautiful and truly useful in real-life application of parametric curves so worth exploring for motivation or simply intriguing fun. See Bezier Curves - Getting the Shape you Want, Dale Hoffman's Contemporary Calculus

rev. 2021-02-27