EXAMPLES XVIII 285 



(8) Find the length of the curve t/ 2 - 4r between <K - and 



i-i. 



(4) Find the area of the surface of revolution produced by that 

 part of the curve t/ 2 = 4# between x = and x - 1, rotating about 

 the axis of x. 



(5) Use the results of Questions 8 and 4 to find the height 

 of the centroid of the arc of the parabola t/ 2 = 4x between x = 

 and x =- 1 above the axis of x. 



(6) Find the height of the centroid of a semicircular arc of 

 radius a above the diameter. 



(7) Find the length of the curve y 2 = O.r 3 between x = 1 and 

 x= 2. 



2 2 



(8) Find the length of the curve X s -f y* = 4 between x = 

 and x = 8. 



(9) Find the area of the surface of revolution produced by 



2 2 



that part of the curve X s f t/ 5 = 4 between x = and # = 8, 

 rotating about the axis of x. What is the height of the centroid 

 of this part of the curve above the axis of x ? 



(10) Find the length of that part of the curve y = 2x x z 

 between x = and x - I. 



(11) Find the length of that part of the curve y = log e x between 

 x = 1 and x = 2. 



(12) Express the equations of the following curves in polar 

 co-ordinates : 



(2) a?y = a*(x* - y z ) 



(3) ay = x*(a z - x 2 ) 



(13) Trace ^ie curve r 2 = 16 sin 2 6+25 cos 2 6, and find the area 

 enclosed by it. 



(14) Transform the equation of the curve (a; 2 + y 2 ) 2 = 9(,r 2 t/ 2 ) 

 into polar co-ordinates. Trace the curve, and find the area of 

 a loop. 



(15) Trace each of the following curves between 6 = and 

 6=27r: 



(1) r6 = 4 



(2) r6^ = 4 



(3) r6 2 = 4 



For each curve find the area of a sector between 6 = and 6 = TC. 



