280 

 For 



also 



PRACTICAL MATHEMATICS 



x = r cos 0, and y = r sin 



2 + y 2 = r~ 



These relations can be used to transform the equation of a 

 curve from one system to the other. 



For example, the equation of a parabola in rectangular co- 

 ordinates is y 2 = 4<ax. 



Then r 2 sin 2 = 4>ar cos 



r = 4a 



cos 



sin 2 



8a cos 

 = 1 - cos 20 



148. The Area of a Curve in Polar Co-ordinates. Let P and Q 

 be two points on a curve, taken very close together. (Fig. 91.) 



JL_l 



FIG. 91, 



The polar co-ordinates of P being r, 0, and of Q (r + Sr), (0 + 80). 

 Let PR be drawn perpendicular to OQ. Then if 80 is small 

 the following relations are approximately true, 



OR = OP = r 

 RQ= Sr 

 and PR = r 80 



Area of the sector OPQ = area of triangle OPR + area of 



triangle RPQ 



= Jr 2 sin 80 + \r 80 Sr 



'2 2 



When 80 is taken to be very small this area becomes - r 2 



2 



