2 1y x

25 2 2

12 1 2 1y dy

21y x 1x y

outerradius

innerradius

thicknessof slice

cylinder

5

14 1 4y dy

5

15 4y dy

52

1

15 4

2y y

25 125 5 4

2 2

25 94

2 2

164

2

8 4 12

If we take a vertical slice and revolve it about the y-axis

we get a cylindrical shell.

2 1y x

Here is another way we could approach this problem:

If we add all of the cylindrical shells together, we can reconstruct the original object.

Imagine cylinders within cylinders in which all you wanted was the volume only of the shell not the inside of the cylinder

If we add all of the cylindrical shells together, we can reconstruct the original object.

If we take a vertical slice and revolve it about the y-axis

we get a cylindrical shell.

cross section

Here is another way we could approach this problem:

2 1y x

cross section

The volume of a thin, hollow cylinder is given by:

=2 thicknessr h r is the x coordinate.

h is the y coordinate.

thickness is dx.r h

thicknesscircumference

2 1y x

C

Lateral surface area of cylinder thicknesscircumference height thickness

r

Picture this cylinder like you would a soup label and imagine it being stretched out flat like this…

?

Notice that your answers were the same with both methods

Another way to look at shells would be to look at the layers of paper formed by the pages of a magazine if you were to roll the magazine up into a cylindrical shape.

Remember that you’re finding the same volume just by different methods; one by rotating horizontal rectangles and the other by rotating vertical ones

2 1y x 2 1y x

cross section

=2 thicknessr h

2=2 1 x x dx

r hthicknesscircumference

If we add all the cylinders from the smallest to the largest:

2 2

02 1 x x dx

2 3

02 x x dx

24 2

0

1 12

4 2x x

2 4 2

12

This is called the shell method because we use cylindrical shells.

2 1y x

2410 16

9y x x

Find the volume generated when this shape is revolved about the y axis.

2410 16

9y x x

If we used horizontal rectangles, we’d get…

R and r are the same curve so this won’t work.

WASHERS…but wait!

Since we can’t solve for x, we can’t use a horizontal slice directly.

Volume of the shell =

2410 16

9y x x

Shell method:

Lateral surface area of cylinder

=circumference height

=2 r h

If we take vertical slicesand revolve them about the y-axiswe get cylinders.

Remember: we don’t want the volume of the cylinder, just the shell of the cylinder

2410 16

9y x x Volume of thin cylinder 2 r h dx

8 2

2

42 10 16

9x x x dx r

h thickness

160

3502.655 cmcircumference

r

h

x

y

2xy Find the volume when rotating this region about the y-axis

2

0

2 )(2 dxxxV

r = xh = y = x2

x

yFind the volume when rotating this region about the x-axis

2xy

4

0)2(2 dyyyV

r = yh = y2

x

y

2xy

Find the volume when rotating this region about the line x = 2

2

0

2 ))(2(2 dxxxV

r = 2 – xh = y = x2

When the strip is parallel to the axis of rotation, use the shell method.

When the strip is perpendicular to the axis of rotation, use the disk or washer method.

When rotating about the y-axis, the radius is x.

y-axisThe height is the distance from the upper curve to the lower curve.

r

h

f (x)

When rotating about the line x = a, the radius is x – a if x > a (if the region is to the right of a)

x = aThe height is the distance from the upper curve to the lower curve.

h

r

When rotating about the line x = a, the radius is a – x if a > x (if the region is to the left of a)

x = a

r

h

The same applies when rotating about the y-axis except that everything needs to be in terms of y

f (x)

f (x)