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.
» Session 80: Parametric Curves
» Session 81: Examples Using Parametrized Curves
» Session 82: Polar Coordinates
» Session 83: Polar Coordinates, Continued
» Session 84: Polar Coordinates and Graphing
» Problem Set 11
In this session you will:
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 to Applications of Integration problems (PDF)
This problem set is from exercises and solutions written by David Jerison and Arthur Mattuck.
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.
Clip 2: Arclength of Parametric Curves
Exploring a Parametric Curve
In this session we compute the arc lengths of some parametrized curves; we also compute the surface area of the solid of revolution formed by an ellipse.
Clip 1: Non-Constant Speed Parametrization
Clip 2: Surface Area of an Ellipsoid
Path of a Falling Object
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.
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
This session describes how to use polar coordinates to find areas. In polar coordinates the basic unit of area is not rectangular; it is a portion of a circle.
Clip 1: Polar Coordinates and Area
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.
Clip 1: Graph of r = 2a cos(θ)
Clip 3: Polar Coordinates and Conic Sections
