Lecture 24 : Volume of solids of revolution by Shell method [Section 24.2]
24.2
Volume of solid of revolution by Shell Method :
In this section we describe another method of finding the volume of solids of revolution.
24.2.1
Definition:
Let be the plane region between the two curves given by functions
which are Riemann integrable and
Suppose that , so that lies on or to the right of the -axis. Then the volume of the solid generated by revolving about the -axis is defined to be . ----------(15)
24.2.2
Note:
(i)
This formula is motivated by the fact that if a thin vertical slice of thickness of the region at a point is
revolved about the -axis, then we get a cylindrical shell of radius and height . Thus, the volume of this thin shell is given by
Hence, the required volume is given by equation (15).
be the plane region between the two curves given by functions 
, so that 
lies on or to the right of the 
-axis. Then the volume of the solid generated by revolving 
about the 
-axis is defined to be 
. ----------(15)
of the region 
at a point 
is 
-axis, then we get a cylindrical shell of radius 
and height 
. Thus, the volume of this thin shell is given by 
