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

2. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Introduction to polar coordinates Polar curves Standard polar curves Applications

3. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Introduction to polar coordinates Polar curves Standard polar curves Applications

4. 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 by the equations:

5. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Introduction to polar coordinates Given that: then:

6. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Introduction to polar coordinates Polar curves Standard polar curves Applications

7. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Introduction to polar coordinates Polar curves Standard polar curves Applications

8. 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

9. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems From the table of values it is then a simple matter to construct the graph of: 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

10. STROUD Worked examples and exercises are in the text 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 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. Polar curves

11. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems As a further example the plot of: r = 2sin 2 θ exhibits the two circles distinctly. Polar curves

12. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Introduction to polar coordinates Polar curves Standard polar curves Applications

13. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Introduction to polar coordinates Polar curves Standard polar curves Applications

14. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Standard polar curves r = a sin θ r = a sin 2 θ

15. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Standard polar curves r = a cos θ r = a cos 2 θ

16. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Standard polar curves r = a sin2θ r = a sin3θ

17. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Standard polar curves r = a cos2θ r = a cos3θ

18. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Standard polar curves r = a(1 + cosθ ) r = a(1 + 2cosθ )

19. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Standard polar curves r 2 = a 2 cos2θ r = aθ

20. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Standard polar curves The graphs of r = a + b cos θ

21. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Introduction to polar coordinates Polar curves Standard polar curves Applications

22. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Introduction to polar coordinates Polar curves Standard polar curves Applications

23. 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

24. 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:

25. STROUD 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 : Applications Volume of rotation of a polar curve

26. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Since: so: Applications Volume of rotation of a polar curve

27. 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:

28. STROUD 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:

29. STROUD Worked examples and exercises are in the text Programme 23: Polar coordinate systems Since: so: Therefore: Applications Surface of rotation of a polar curve

30. STROUD Worked examples and exercises are in the text Programme 23: 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