Polar Coordinates

1
Polar Coordinates

• Definition Conversions and Integration

2
Where is it
Coordinate systems are used to locate the
position of a point.

• In polar coordinates
• We break up the plane with circles centered at
  the origin and with rays emanating from the
  origin.
• We locate a point as the intersection of a circle
  and a ray.

• In rectangular coordinates
• We break up the plane into a grid of horizontal
  and vertical line lines.
• We locate a point by identifying it as the
  intersection of a vertical and a horizontal line.

3
Locating points in Polar Coordinates
The first coordinate r 2 indicates the
distance of the point from the origin.
The second coordinate /6 indicates the
distance counter-clockwise around from the
positive x-axis.
4
Locating points in Polar Coordinates
Note however that every point in the plane as
infinitely many polar representations.
5
Locating points in Polar Coordinates
Note however that every point in the plane as
infinitely many polar representations.
6
Locating points in Polar Coordinates
Note however that every point in the plane as
infinitely many polar representations.
And we can go clockwise or counterclockwise
around the circle as many times as we wish!
7
Converting Between Polar and Rectangular
Coordinates
It is fairly easy to see that if (xy) and (r q)
represent the same point in the plane
These relationships allow us to convert back and
forth between rectangular and polar coordinates
8
Integration in Polar Coordinates

• Non-rectangular Integration Elements

9
Small Changes in r and q
Suppose we consider a small change from r . . .
to r dr
10
Small Changes in r and q
Suppose we consider a small change from r . . .
to r dr This gives us a thin ring around
the origin.
11
Small Changes in r and q
Suppose we consider a small change from q . . .
to q dq
12
Small Changes in r and q
Suppose we consider a small change from q . . .
to q dq This gives us a pie-shaped wedge
that is subtended by the angle dq.
13
Small Changes in r and q
Intersecting the thin ring . . . and the
pie-shaped wedge . . . we get . . .
14
Small Changes in r and q
Intersecting the thin ring . . . and the
pie-shaped wedge . . . we get . . .
15
Small Changes in r and q
In order to integrate a function given in polar
coordinates (without first converting to
rectangular coordinates!) we need to know the
area of this little piece.
dr
Why
dq
16
Integration in Polar Coordinates
In order to integrate a function given in polar
coordinates we will first chop up our region
into a bunch of concentric circles and rays
emanating from the origin.
(rq f(rq))
(rq)
Now do this for each little wedge and add up
the volumes of the towers.
Problem the volume of the tower is the area
of the base times the height. But the base is
not a rectangle so its area is not dr dq!
17
Area of the Small Bit
A area of sector of a circle
q
18
Area of a Small Bit
In order to integrate a function given in polar
coordinates (without first converting to
rectangular coordinates!) we need to know the
area of this little piece.
dr
r
dq
19
Area of a Small Bit