http://www.ck12.org Chapter 5. Applications of Definite Integrals
Figure 14
Notice however that(r 2 −r 1 )is the thickness of the shell and^12 (r 2 +r 1 )is the average radius of the shell.
Thus
V= 2 π·[average radius]·[height]·[thickness].Replacing the average radius with a single variablerand usinghfor the height, we have
V= 2 π·r·h·[thickness].In general the shell’s thickness will bedxordydepending on the axis of revolution. This discussion leads to the
following formulas for rotation about an axis. We will then use this formula to compute the volumeVof the solid
of revolution that is generated by revolving the region about thex−axis.
Volume By Cylindrical Shell about they−Axis
Supposefis a continuous function in the interval[a,b]and the regionRis bounded above byy=f(x)and below
by thex−axis, and on the sides by the linesx=aandx=b.IfRis rotated around they−axis, then the cylinders are
vertical, withr=xandh=f(x).The volume of the solid is given by
V=
∫b
a^2 πrhdx=∫a
b^2 πx f(x)dx.