1. inner
radius
cylinder
outer
radius
thickness
of slice

2. Here is another way we could approach this problem:
If we take a vertical slice
and revolve it about the y-axis
we get a cylindrical shell.
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

3. Here is another way we could approach this problem:
cross section
If we take a vertical slice
and revolve it about the y-axis
we get a cylindrical shell.
If we add all of the cylindrical shells together, we can reconstruct the original object.

4. r is the x coordinate. h is the y coordinate.
Picture this cylinder like you would a soup label and imagine it being stretched out flat like this…
cross section
The volume of a thin, hollow cylinder is given by: ? C
r is the x coordinate.
h is the y coordinate.
thickness is dx.

5. Notice that your answers were the same with both methods
Remember that you’re finding the same volume just by different methods; one by rotating horizontal rectangles and the other by rotating vertical ones
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.

6. This is called the shell method because we use cylindrical shells.
cross section
If we add all the cylinders from the smallest to the largest:

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

8. If we used horizontal rectangles, we’d get…
WASHERS…but wait!
R and r are the same curve so this won’t work.
Since we can’t solve for x, we can’t use a horizontal slice directly.

9. If we take vertical slices Shell method:
and revolve them about the y-axis
we get cylinders.
Volume of the shell =
Remember: we don’t want the volume of the cylinder, just the shell of the cylinder

11. Find the volume when rotating this region about the y-axis
y = x2

12. Find the volume when rotating this region about the x-axisy
h =

13. Find the volume when rotating this region about the line x = 2
y = x2

14. The height is the distance from the upper curve to the lower curve.
When the strip is perpendicular to the axis of rotation, use the disk or washer method.
When the strip is parallel to the axis of rotation, use the shell method.
y-axis
The height is the distance from the upper curve to the lower curve.
f (x)
When rotating about the y-axis, the radius is x. h r

15. x = a
The height is the distance from the upper curve to the lower curve.
f (x) h
When rotating about the line x = a, the radius is a – x if a > x (if the region is to the left of a) r
x = a
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) h
The same applies when rotating about the y-axis except that everything needs to be in terms of y r p