When you start a problem, you may not know which method is best. Just pick one and see what happens!

- If you decide to use vertical slices:
- then you must integrate with respect to x and your integral will have a dx in it.
- Your integral will be ,
where a = the x-value of the leftmost slice, and
b = the x-value of the rightmost slice. For each x-value, you'll
get a vertical slice whose area A(x) is
a function of x. Here's an example when the volume
is obtained by revolving a plane region around a horizontal axis
(i.e., the method of vertical washers):

- If you decide to use horizontal slices:
- then you must integrate with respect to y and your integral will have a dy in it.
- Your integral will be ,
where c = the y-value of the bottom slice, and
d = the y-value of the top slice. For each y-value, you'll
get a horizontal slice whose level is y, and A(y), the area of
the slice, is a function of y. Here's an example when the volume
is obtained by revolving a plane region around a vertical axis
(i.e., the method of horizontal washers).
**Note that you have to invert the functions y=f(x) and y=g(x) to express x as functions of y. That is a disadvantage to using horizontal slices for figures of revolution revolved around a vertical axis.**

- Now suppose you have a problem like that in Figure 1, but you want
to use concentric shells. Then the integral you must use is
.
Note that this changes the variable of integration, compared to
the method shown in Figure 1,
**but you have to invert the functions y=f(x) and y=g(x) to express x as functions of y. That is a disadvantage to using shells for figures of revolution revolved around a horizontal axis.** - And to use concentric shells with a problem like that in Figure 2,
the integral you must use is
.
Note again that this changes the variable of integration, compared to
the method shown in Figure 2;
**using shells for figures of revolution revolved around a vertical axis of revolution lets you avoid inverting the functions y=f(x) and y=g(x). This can be a big advantage!** - So why would you ever do a Figure 1 problem with shells, or a Figure 2 problem with washers, since then you have to invert y=f(x) and y=g(x)? See example 3.3 (p. 429); this one has a horizontal axis of revolution so you'd like to do it like in Figure 1 above, but if you do you end up needing to break it into two integrals (this is because the function at the top of the plane region changes from y=x to y=2-x). If you use shells, you end up needing to invert y=f(x) and y=g(x) (which is a disadvantage), but you need only one integral (which is an advantage).