Question

Refer to the figure and find the volume V generated by rotating the given region about the specified line.

Answer 1

Volume generated by the areas R2 + R3 around the line OC
= (1/3) pi 1^2 * 3  =  pi sq units

Volume generated by area R2 around OC  is found as follows

the  equation of the curve can be written as  x = y^4 / 81

Integral between limits y =  0 and 3  of  the curve is  

pi INT  x^2 dy    = INT pi  y^8 / 81^2  dx   

= pi [3^9 / 9*81^2] 

= pi/3

So the volume generated by R3 about OC = pi - pi/3  = 2pi/3