STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems POLAR COORDINATE SYSTEMS PROGRAMME 23

STROUD

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

POLAR COORDINATE

SYSTEMS

PROGRAMME 23

STROUD



Introduction to polar coordinates

Polar curves

Standard polar curves

Applications

STROUD




Polar curves


Applications

STROUD




The position of a point in a plane can be represented by:

(a) Cartesian coordinates (x, y)(b) polar coordinates (r, θ)

The two systems are related bythe equations:

cos and sinx r y r

STROUD




Given that:

then:cos and sinx r y r

2 2 1 and tan yr x yx

STROUD




Polar curves


Applications

STROUD




Polar curves


Applications

STROUD



In polar coordinates the equation of a curve is given by an equation of the form r = f (θ ) whose graph can be plotted in a similar way to that of an equation in Cartesian coordinates. For example, to plot the graph of:

r = 2sin θ between the values 0 ≤ θ ≤ 2π

a table of values is constructed:

Polar curves

STROUD



From the table of values it is then asimple matter to construct the graphof: r = 2sin θ

(a) Choose a linear scale for r and indicate it along the initial line.

(b) The value for r is then laid off along each direction in turn, point plotted, and finally joined up with a smooth curve.

Polar curves

STROUD



Note: When dealing with the 210º direction, the value of r obtained is negative and this distance is, therefore, laid off in the reverse direction which brings the plot to the 30º direction.

For values of θ between 180º and 360ºthe value obtained for r is negative and the first circle is retraced exactly. The graph, therefore, looks like one circle but consists of two circles one on top of the other.

Polar curves

STROUD



As a further example the plot of:

r = 2sin2θ

exhibits the two circles distinctly.

Polar curves

STROUD




Polar curves


Applications

STROUD




Polar curves


Applications

STROUD




r = a sin θ r = a sin2θ

STROUD




r = a cos θ r = a cos2θ

STROUD




r = a sin2θ r = a sin3θ

STROUD




r = a cos2θ r = a cos3θ

STROUD




r = a(1 + cosθ ) r = a(1 + 2cosθ )

STROUD




r2 = a2cos2θ r = aθ

STROUD




The graphs of r = a + b cos θ

(a) (cardioid)

(b) (re-entrant loop)

(b) (no cusp or re-entrant loop)

a b

a b

a b

STROUD




Polar curves


Applications

STROUD




Polar curves


Applications

STROUD



Applications

Area of a plane figure bounded by a polar curve

Volume of rotation of a polar curve

Arc length of a polar curve

Surface of rotation of a polar curve

STROUD



Applications

Area of a plane figure bounded by a polar curve

Area of sector OPQ is δA where:

Therefore:

1 ( )sin2

A r r r

2

1

0

0

22

1 sin( )2

1 so that 2 2

A dALimd

Lim r r r

r A r d

STROUD



Area of sector OPQ is δA where:

The volume generated when OPQ rotates about the x-axis is δV where :

1 ( )sin2

A r r r

area OPQ distance travelled by its centroid (Pappus)1 2( )sin .2 . sin2 3

V

r r r r

Applications


STROUD



Since:

so:

1 2( )sin .2 . sin2 3

V r r r r

2

1

0

2

0

3 3

2 sin( ) sin3

2 2 and so sin3 3

V dVLimd

Lim r r r

r V r d

Applications


STROUD



Applications

Arc length of a polar curve

By Pythagoras:

so that:

therefore:

22 2 2.s r r

2 22s rr

2

1

2 22 2 Hence ds dr drr s r d

d d d

STROUD



Applications


If the element of arc PQ rotates about the x-axis then, by Pappus’ theorem, the area of the surface generated is given as:

S = (the length of the arc) × (the distance travelled by its centroid)

That is:

.2 .PL

.2 sinS s

s r

STROUD



Since:

so:

Therefore:

.2 .PL

.2 sinS s

s r

.2 sinS s r

2

1

2 22 22 sin and so 2 sindS dr drr r S r r d

d d d

Applications


STROUD


Programme 23: Polar coordinate systemsLearning outcomes

Convert expressions from Cartesian coordinates to polar coordinates

Plot the graphs of polar curves

Recognize equations of standard polar curves

Evaluate the areas enclosed by polar curves

Evaluate the volumes of revolution generated by polar curves

Evaluate the lengths of polar curves

Evaluate the surfaces of revolution generated by polar curves

Documents

STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems POLAR COORDINATE SYSTEMS PROGRAMME 23