AREAS OF SURFACES OF REVOLUTION 275 



Example. To find the area of the curved surface of a spherical 

 cap of height h and radius of base r. 



Let a be the radius of the sphere. The whole surface of the 

 sphere can be produced by the semicircle rotating about the 

 vertical diameter or the axis OY. The surface of the spherical 

 cap is produced by the arc AC rotating about the axis OY. 



Choosing the centre of the circle as origin, the equation to the 

 circle is x 2 + t/ 2 = a 2 , while the limits of y for the arc AC will 

 be y = a h and y = a. 



Since x z + v* = a 2 



Then 



and 



dii 



2x + 2y -f- = 



Now 



SOY 



27r 

 27ta 



f a, 

 l x-dy 



Ja-h X 



| dif 



Ja-h 



[a (a h)] 



