**Chapter 16**

August 17 2017

**Chapter 16 Polar Coordinates**
**16.1 The Polar Coordinate System**
- a point is located by means of its distance and direction from the origin
- polar coordinates (r, angle)
- direction angle = 0, the positive x-axis, is called the polar axis
- it is important to know the connection between rectangular and polar coordinates
- x = r cos angle, y = r sin angle
- just as in the case of rectangular coordinates
- the graph of a polar equation
- F (r, angle) = 0
- is the set of all points P = (r, angle) whose polar coordinates satisfies the equation
- in most situations we will encounter
- the equation can be solved for r, and takes the form
- r = f(angle)
- we merely choose a convenient sequence of values for angle
- each determining its own direction from the origin
- and compute the corresponding values of r that tell us how far to go in each of these directions
**16.2 More Graphs of Polar Equations**
- the curve
- r = a (1 + cos angle) with a > 0
- is a cardioid
- the curve r = a (1 + 2 cos angle) with a > 0
- is a limacon (snail)
- r2 = 2a2 cos 2 angle is a lemniscate (figure eight)
**16.3 Polar Equations of Circles, Conics, and Spirals**
- we had translated polar coordinates directly to rectangular coordinates
- another way is to derive the polar coordinates from some characteristic geometric property of the curve
- polar coordinates are particularly well suited for working with conic sections
**16.4 Arc Length and Tangent Lines**
- we can use integration to find the length of arc lengths in polar coordiantes
- the formula for ds can also be used to find areas of the surfaces of revolution
**16.5 Areas in Polar Coordinates**
- we think of the total area A as a result of adding up elements of area dA