How in general to set up your integral for computing volume

There are three main methods you can use: horizontal slices; vertical slices; or concentric shells. (But concentric shells work only if you have a volume of revolution, whereas you can always try the method of slicing. For a figure of revolution, the slices are perpendicular to the axis of revolution, so use vertical slices for a horizontal axis of revolution, and horizontal slices for a vertical axis of revolution.)

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.

Concentric Shells

• 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).