7.3 - Volumes by Cylindrical Shells
A little about Cylindrical Shells. This would be a solid that is enclosed by two cylinders.
Basically, think of a beer insulator: something you put a can of whatever in, and it’s open at the
top so that you can still drink from the can. But, it’s still in the shape of the can itself.
Essentially, it looks like a cylinder. I guess think of a can of soup too, but that doesn’t have much
thickness to it. Okay, so, anyway, how do we ▯nd the volume of such a gadget?
Recall that cylinders are made up of a bunch of circles, and in fact, the volume of a cylinder would
be ▯r h. But this is not a cylinder we’re talking about. We want the volume of that stu▯ between
the two cylinders (that thickness essentially). Well, let’s start with the circles. There are two radii
to this cylindrical shell, the inside radius (r ) and the outside radius (r ). And then, of course,
there’s the height we shall call h. So, the volume of this guy is
V = [area of cross section] ▯ [height]
= [▯r ▯2▯r ]h1
= ▯(r + r )(r ▯ r )h
▯ 1 2 ▯ 1
= 2▯ (r1+ r 2 ▯ h ▯ (r2▯ r 1
And that bit of algebra at the end there is to help you see a little better what’s happening.
2(r1+ r 2 is the average radius of the cylinder, height is of course h, and then r 2 r i1 the
V = 2▯ ▯ [avg radius] ▯ [height] ▯ [thickness]
So, now that we’ve explained a little about what cylindrical shells are, we must go on and say why
we need to know about them. Well, in 7.1, when you ▯nd the volume of revolution by disks or
washers, your thickness must be perpendicular to the axis of revolution. Ie, if you’re revolving
about the x-axis, you’re using dx, and if it’s over the y-axis, you’re using dy. Cylindrical shells
form when you go parallel to the axis of revolution. That is to say, if you revolve a region about
the y-axis, you better make sure that everything is in terms of x! And the formula for revolving
about the y-axis using shells is
V = 2▯xf(x) dx
Look familiar? x is the average radius, f(x) is the height, and dx is the thickness of these
cylindrical shells. And since you want all these shells together to ▯nd the entire volume, you
If you want to ▯nd the volume of a region revolved about the x-axis, make sure that everything is
in terms of y!
Example: Using cylindrical shells, ▯nd the volume that results when the area enclosed byxy =
and y = x is revolved about the x-axis.
Okay, so x-axis using cylindrical shells means everything must be in terms of y. So, solving for x
in both of those curves, we get x = y and x =y (we only want the positive root). Setting these
two equal to each other, we also get y = 0 and y = 1. So, our volume turns out to be
Z 1 p
V = 2▯ y( y ▯ y ) dy
= 2▯ y3=2▯ y dy
▯ ▯ 1
2 5=2 1 4
= 2▯ y ▯ y
▯ 5 ▯ 4 0
= 2▯ ▯
Thankfully, this is the exact volume that was in the notes yesterday, only we used the washer
method. This brings me to a certain point: you can use either method - dish/washers or
cylindrical shells - so long as you use them correctly. Both will yield you the same answer. Just
remember: disks and washers your variable will be the same in which you are revolving around (ie
revolve around x-axis, your variable will be x), which shells will be the opposite variable (ie,
revolve around x-axis means you will integrate with respect to y). Although they will both yield
the same answer, occasionally you may run into a problem where doing it one way over the other
is much much easier, or it may even be impossible to do it the other way. Also, while using either
method gives you the same answer, the same cannot be said about comparing revolving around
the y-axis to the x-axis. Treat them di▯erently, for they will give you di▯erent answers.
On your exam, if you are asked to ▯nd the volume over say the x-axis, but it says nothing as to
how you are to ▯nd it, then you may use either method (make sure you revolve it about the axis it
says though!). However, if you are speci▯cally asked to solve it a certain way, you must do it the
way it asks.