18.01SC | Fall 2010 | Undergraduate

# Single Variable Calculus

4. Techniques of Integration

## Part C: Parametric Equations and Polar Coordinates

« Previous | Next »

We start by extending our technique for finding the length of a portion of a graph to cover any curve we can describe algebraically. The remainder of this part of the course explores Polar Coordinates, an alternative method of describing locations in the plane.

« Previous | Next »

« Previous | Next »

### Overview

In this session you will:

• Do practice problems
• Use the solutions to check your work

### Problem Set

Use Applications of Integration (PDF) to do the problems below.

Section Topic Exercises
4E Parametric equations 2, 3, 8
4F Arclength 1d, 4, 5, 8
4G Surface area 2, 5

#### Solutions

Solutions to Applications of Integration problems (PDF)

This problem set is from exercises and solutions written by David Jerison and Arthur Mattuck.

« Previous | Next »

« Previous | Next »

### Overview

To study curves which aren’t graphs of functions we may parametrize them, identifying a point (x(t), y(t)) that traces a curved path as the value of t changes. We can then use our technique for computing arclength, differential notation, and the chain rule to calculate the length of the parametrized curve over the range of t.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Parametric Curve

Clip 2: Arclength of Parametric Curves

### Worked Example

Exploring a Parametric Curve

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Remarks on Notation

« Previous | Next »

« Previous | Next »

### Overview

Like Cartesian coordinates, polar coordinates are used to identify the locations of points in the plane. The polar coordinate system will be useful for many problems you encounter at MIT, such as those involving circular motion or radial forces.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Introduction to Polar Coordinates

Clip 2: Simple Examples in Polar Coordinates

Clip 3: Translating y = 1 into Polar Coordinates

Clip 4: Equation of an Off-Center Circle

### Recitation Video

#### Converting From Polar to Rectangular Coordinates

« Previous | Next »

« Previous | Next »

### Overview

Graphing curves described by equations in polar coordinates can be very rewarding, but we must be attentive when plotting points whose radii are negative. Occasionally it is helpful to convert from polar coordinates to Cartesian (xy) coordinates in order to better understand a curve.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Graph of r = 2a cos(θ)

Clip 2: Graph of r = sin(2θ)

Clip 3: Polar Coordinates and Conic Sections

### Recitation Video

#### Graph of r = 1 + cos (θ/2)

« Previous | Next »

## Course Info

Fall 2010
##### Learning Resource Types
Exams with Solutions
Lecture Notes
Lecture Videos
Problem Sets with Solutions
Simulations
Recitation Videos