196 



Since y = cx n 



dy = ncx n ~ l dx 

 and x dy = ncx n dx 



Therefore area B 



J 

 rb 



dx 



= \xdy 

 = nc I x n 



Ja 



nc 



n + 



nc 



n + 1 



A comparison of the results for the areas A and B shows that 

 if the law of the curve is of the form y = cx n , then the area B is 

 n times the area A. 



112. Example 1. Working between the limits x = 2 and x = 3 



for the curve y = Sx?. Find (1) the area bounded by the ordinates 

 at x = 2 and x = 3, and (2) the area bounded by the abscissae 

 which correspond to the ordinates at x = 2 and x = 3. 



f 3 

 Then area (1) = | ?/ dx 



r dx 



T 



= ,, ,,J 2 



= l-2{9"V/3-4V'2} 

 = 11-92 



and area (2) = l# dy 



9 f 3 ^ j 9 i j 

 =-- -\ x* dx, since CM/ = -ara dx 



2. Jz A 



= d2 



5L _ia 



= l-8{9V3-4\/2} 

 = 17-88 



