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christopher-todd
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One way to give someone directions is to tell them to go three blocks East and five blocks South.
Another way to give directions is to point and say “Go a half mile in that direction.”
Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle.
Initial ray
r A polar coordinate pair
determines the location of a point.
,r
r – the directed distance from the origin to a point
Ө – the directed angle from the initial ray (x-axis) to ray OP.
1 2 02
r
r a
o
(Circle centered at the origin)
(Line through the origin)
Some curves are easier to describe with polar coordinates:
(Ex.: r = 2 is a circle of radius 2 centered around the origin)
(Ex. Ө = π/3 is a line 60 degrees above the x-axis extending in both directions)
30o
2
More than one coordinate pair can refer to the same point.
2,30o
2,210o
2, 150o
210o
150o
All of the polar coordinates of this point are:
2,30 360
2, 150 360 0, 1, 2 ...
o o
o o
n
n n
Each point can be coordinatized by an infinite number of polar ordered pairs.
To find the slope of a polar curve:
dy
dy ddxdxd
sin
cos
dr
ddr
d
sin cos
cos sin
r r
r r
We use the product rule here.
A lot like parametric slope.
Example: 1 cosr sinr
sin sin 1 cos cosSlope
sin cos 1 cos sin
2 2sin cos cos
sin cos sin sin cos
2 2sin cos cos
2sin cos sin
cos 2 cos
sin 2 sin
The length of an arc (in a circle) is given by r. when is given in radians.
Area Inside a Polar Graph:
For a very small , the curve could be approximated by a straight line and the area could be found using the triangle formula: 1
2A bh
r dr
21 1
2 2dA rd r r d
Example: Find the area enclosed by: 2 1 cosr
2 2
0
1
2r d
2 2
0
14 1 cos
2d
2 2
02 1 2cos cos d
2
0
1 cos 22 4cos 2
2d
This graph is called a limaƈon.
Notes:
To find the area between curves, subtract:
2 21
2A R r d
Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.
When finding area, negative values of r cancel out:
2sin 2r
22
0
14 2sin 2
2A d
Area of one leaf times 4:
2A
Area of four leaves:
2 2
0
12sin 2
2A d
2A
To find the length of a curve:
Remember: 2 2ds dx dy
Again, for polar graphs: cos sinx r y r
If we find derivatives and plug them into the formula, we (eventually) get:
22 dr
ds r dd
So: 22Length
drr d
d