1. POLAR COORDINATE SYSTEMS
PROGRAMME 22
POLAR COORDINATE SYSTEMS

2. Programme 22: Polar coordinate systems
Introduction to polar coordinates
Polar curves
Standard polar curves
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

3. Programme 22: Polar coordinate systems
Introduction to polar coordinates
Polar curves
Standard polar curves
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

4. Programme 22: Polar coordinate systems
Introduction to polar coordinates
The position of a point in a plane can be represented by:
Cartesian coordinates (x, y)
polar coordinates (r, )
The two systems are related by
the equations:

5. Programme 22: Polar coordinate systems
Introduction to polar coordinates
Given that:
then:

6. Programme 22: Polar coordinate systems
Introduction to polar coordinates
Polar curves
Standard polar curves
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

7. Programme 22: Polar coordinate systems
Polar curves
In polar coordinates the equation of a curve is given by an equation of the form r = f ( ) 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  between the values 0    2
a table of values is constructed:

8. Programme 22: Polar coordinate systems
Polar curves
From the table of values it is then a
simple matter to construct the graph
of:
r = 2sin 
Choose a linear scale for r and indicate it along the initial line.
The value for r is then laid off along each direction in turn, point plotted, and finally joined up with a smooth curve.

9. Programme 22: Polar coordinate systems
Polar curves
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  between 180º and 360º
the 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.

10. Programme 22: Polar coordinate systems
Polar curves
As a further example the plot of:
r = 2sin2
exhibits the two circles distinctly.

11. Programme 22: Polar coordinate systems
Introduction to polar coordinates
Polar curves
Standard polar curves
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

12. Programme 22: Polar coordinate systems
Standard polar curves
r = a sin  r = a sin2

13. Programme 22: Polar coordinate systems
Standard polar curves
r = a cos  r = a cos2

14. Programme 22: Polar coordinate systems
Standard polar curves
r = a sin2 r = a sin3

15. Programme 22: Polar coordinate systems
Standard polar curves
r = a cos2 r = a cos3

16. Programme 22: Polar coordinate systems
Standard polar curves
r = a(1 + cos ) r = a(1 + 2cos )

17. Programme 22: Polar coordinate systems
Standard polar curves
r2 = a2cos2 r = a

18. Programme 22: Polar coordinate systems
Standard polar curves
The graphs of r = a + b cos 

19. Programme 22: Polar coordinate systems
Introduction to polar coordinates
Polar curves
Standard polar curves
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

20. Programme 22: Polar coordinate systems
Area of a plane figure bounded by a polar curve
Area of sector OPQ is A where:
Therefore:

21. Programme 22: Polar coordinate systems
Introduction to polar coordinates
Polar curves
Standard polar curves
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

22. Programme 22: Polar coordinate systems
Volume of rotation of a polar curve
Area of sector OPQ is A where:
The volume generated when OPQ rotates about the x-axis is V where :

23. Programme 22: Polar coordinate systems
Volume of rotation of a polar curve
Since:
so:

24. Programme 22: Polar coordinate systems
Introduction to polar coordinates
Polar curves
Standard polar curves
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

25. Programme 22: Polar coordinate systems
Arc length of a polar curve
By Pythagoras:
so that:
therefore:

26. Programme 22: Polar coordinate systems
Introduction to polar coordinates
Polar curves
Standard polar curves
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

27. Programme 22: Polar coordinate systems
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:

28. Programme 22: Polar coordinate systems
Surface of rotation of a polar curve
Since:
so:
Therefore:

29. Programme 22: Polar coordinate systems
Learning 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