Polar Coordinates - Calculus II - Lecture Slides, Slides of Calculus

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9.3 Polar Coordinates

9.4 Areas and Lengths in Polar

Coordinates

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.

θ Initial ray

r A polar coordinate pair

determines the location of

a point.

( r ,^ θ )

Polar Coordinates

30

o

2

More than one coordinate pair can refer to the same point.

( 2,30 )

o

( 2, 210 )

o = −

( 2,^150 )

o = − −

210

o

150

o −

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

• ⋅

− − + ⋅ = ± ±

The connection between

Polar and Cartesian coordinates

θ

r

( x , y )

y

x

From the right angle triangle in the picture one immediately gets the following correspondence between the Cartesian Coordinates ( x,y ) and the Polar Coordinates ( r,θ ) assuming the Pole of the Polar Coordinates is the Origin of the Cartesian Coordinates and the Polar Axis is the positive x -axis.

x = r cos( θ ) y = r sin( θ )

r^2 = x^2 + y^2 tan( θ) = y/x

Using these equations one can easily switch between the Cartesian and the Polar Coordinates.

Symmetry with respect to x-axis: If (r, θ) is on the graph,

0

1

1 2

r = 2 cos θ^ r

r

so is (r, -θ).

Examples of Polar Curves

Symmetry with respect to y-axis: If (r, θ) is on the graph,

r

r = 2sin θ

r

so is (r, π-θ)

0

1

2

-1 1

or (-r, -θ).

Examples of Polar Curves

2sin 2.15( )

0 16

r θ

θ π

=

≤ ≤

Examples of Polar Curves

dA = r d θ

(^12)

2

A r d

= θ ∫

Areas in Polar Coordinates

2

0

1 cos 2 2 4 cos 2 2

d

π θ θ θ

= + + ⋅ ∫

2

0

3 4 cos cos 2 d

π = + θ + θ θ ∫

2

0

1 3 4sin sin 2 2

π

= θ + θ + θ

= 6 π − 0

= 6 π

Remember:

For polar graphs: (^) x = r cos θ y = r sinθ

If we find derivatives and plug them into the formula, we

(eventually) get:

2 2 Length

dr r d d

β

α

  = +    

Arc Length in Polar Coordinates

L =

dx

( ) dt

2

dy

( ) dt

2

α

β

∫ dt