POLAR COORDINATE SYSTEMS PROGRAMME 22 POLAR COORDINATE SYSTEMS

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

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

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:

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

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

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:

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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 :

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

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

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

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

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:

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

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