Volume using The Shell Method

The shell method is like the washer method only the rotation is about the y-axis.

http://www.rkm.com.au/animations/animation-solid-revolution-2.html

If we take a vertical slice and revolve it about the y-axis we get cylinders. Then we add up the volume of all the cylinders

Sum of the cylindrical shells

Volume = ∑circumference ∙ height ∙thickness

n

dxhrn

v1

2lim

One Function:

The volume of the solid obtained by revolving R about the y-axis using cylindrical shells is given by:

b

a

dxxfxV )(2

1. Find the volume of the solid obtained by revolving f(x) about the y-axis if

4024)( 2 xforxxf

Two Functions:

The volume of the solid of revolution of the region bounded by f(x) and g(x) is given by:

b

a

dxxgxfxV )()(2

2. Find the volume of the revolution of the region bounded by the curves

axisyandxyandxy 422

3. Find the volume of the solid of revolution bounded by

rotated about the y-axis

13)(3)( 2 xxgandxxf

10 x

Practice 7-3 Shells

Reference

http://www.msstate.edu/dept/abelc/math/volume.html