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

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

Introduction to polar coordinates

Polar curves

Standard polar curves

Applications

STROUD

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Introduction to polar coordinates

Polar curves

Standard polar curves

Applications

STROUD

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Introduction to polar coordinates

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

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Introduction to polar coordinates

Given that:

then:cos and sinx r y r

2 2 1 and tan yr x yx

STROUD

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Introduction to polar coordinates

Polar curves

Standard polar curves

Applications

STROUD

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Introduction to polar coordinates

Polar curves

Standard polar curves

Applications

STROUD

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

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

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Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

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

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Programme 23: Polar coordinate systems

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

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Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

As a further example the plot of:

r = 2sin2θ

exhibits the two circles distinctly.

Polar curves

STROUD

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Introduction to polar coordinates

Polar curves

Standard polar curves

Applications

STROUD

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Introduction to polar coordinates

Polar curves

Standard polar curves

Applications

STROUD

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Standard polar curves

r = a sin θ r = a sin2θ

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Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Standard polar curves

r = a cos θ r = a cos2θ

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Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Standard polar curves

r = a sin2θ r = a sin3θ

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Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Standard polar curves

r = a cos2θ r = a cos3θ

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Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Standard polar curves

r = a(1 + cosθ ) r = a(1 + 2cosθ )

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Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Standard polar curves

r2 = a2cos2θ r = aθ

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Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Standard polar curves

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

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Introduction to polar coordinates

Polar curves

Standard polar curves

Applications

STROUD

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Introduction to polar coordinates

Polar curves

Standard polar curves

Applications

STROUD

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

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

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

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

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Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

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

Volume of rotation of a polar curve

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Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

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

Volume of rotation of a polar curve

STROUD

Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

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

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Worked examples and exercises are in the text

Programme 23: Polar coordinate systems

Applications

Surface of rotation of a polar curve

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

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Programme 23: Polar coordinate systems

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

Surface of rotation of a polar curve

STROUD

Worked examples and exercises are in the text

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