Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

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

Overview

Some calculus operations can be simplified by working with equation in polar form rather than rectangular form. All the calculus we have done with rectangular equations we'll redo in this lesson but using polar form. We'll find derivatives and integrals of polar equations, slopes of tangent lines to polar curves, arc lengths of polar curves, and more.

Objectives

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

• 10.3.1 Find the first derivative of a polar equation
• 10.3.2 Find the slope and equation of a tangent line to a polar curve
• 10.3.3 Find the point(s) at which the tangent line to a polar curve is horizontal or vertical
• 10.3.4 Given the graph of a polar curve identify its critical points, inflection points, relative extrema, and concavity
• 10.3.5 Set-up and evaluate the integral that gives the arc length of a polar curve
• 10.3.6 Set-up and evaluate the integral that gives the area under (or inside) a polar curve
• 10.3.7 Set-up and evaluate the integral that gives the area between two polar curve
• 10.3.8 Use integration to find the volume and surface area of a solid formed by revolving a polar curve about an axis

Terminology & Formulae

Be sure you are able to properly use each of these three formulas with polar equations.

First Derivative of a Polar Equation:  dy/dx= (r*costheta+(dr)/(d theta)*sintheta)/(-r*sintheta+(dr)/(d theta)*costheta)

Arc Length of a Polar Curve:  "AL"= int_alpha^beta sqrt(r^2 + ((dr)/(d theta))^2) d theta

Area in a Polar Curve:  "Area" = 1/2 int_alpha^beta r^2 d theta

Mini-Lectures and Examples

Supplemental Resources (recommended)

Simple sheet with Calculus Formulas for Polar Equations from Dr. Rayman, Univ. of N. Florida.

Supplemental Resources (optional)

rev. 2021-03-06