How To Find Volumes Using the Shell Method (in Full)

Video Series: Applications of Integration (Complete In-Depth Tutorials For Calculus)

(D4M) — Here is the video transcript for this video.

00:00
in this episode you’ll learn how to find the volume of a solid of revolution
using the shell method step by step by step let’s do some math [Music]
hi everyone welcome back we’re going to begin uh talking about the shell method
by looking at this example here and so what i’m going to do is start off
by sketching this graph right here and in fact let’s shade in this region and
we’re going to revolve around the y-axis here so let’s look at a sketch here so
this is going to go through the origin here it’s going to come up and then it’s
going to just fall down it crosses right here at 1.
and so let’s go ahead and call this function f and let’s say it’s x minus x
to the third there and then now let’s go ahead and shade in
this region right here this is the interest this region right here that
we’re interested in and we’re going to revolve this right here around the

00:01
the y-axis there so i usually like to shade them in and look at them right there
all right and so yeah we’re going to revolve this around the
the y-axis here so let’s put a y-axis up here and
x-axis good so this is the region right here that we’re going to revolve around
the y-axis and we do that we’re going to
create a solid and we’re going to try to find the volume so find the volume of
the solid of revolution now if you don’t know what a solid of revolution is
and you haven’t been following along in the series then let me just take a quick
second here to mention that this episode is part of the series applications of
integration complete in-depth tutorials for calculus so in the previous episodes
we talked about finding volumes of solids of revolution in particular we
talked about the disk method and the washer method
and so in today’s episode we’re going to talk about the gel method now

00:02
i’m going to pick a representative right here so you might
remember from calculus 1 to find the area under the graph
you pick uh rectangles you pick uh representatives and then you add up a
whole bunch of tiny little representatives here
and so i’m going to call this right here this distance right here this little
tiny distance of delta x and we have the uh function f and we got an x
tick mark ticked off right there in x and so we can go and
try to find the volume of the solid that’s generated and so the way we’re
going to do that is realize that when we revolve around the y-axis
this rectangle here is going to become a cylindrical shell so let me try to
sketch the graph over here to see what that would look like so i’m gonna have a

00:03
cylindrical shell and it’s gonna come down here like this and
that shell is gonna have some kind of thickness to it so
we can sketch the thickness i guess in blue just to represent that rectangle
right there so whenever we revolve this around the y
axis here that that that rectangle is going to get revolved around the y axis
the whole thing gets revolved but i’m just looking at this representative here
now how do we find the volume of a cylindrical shell here so let’s call
this a cylindrical shell and just shell method for short
but we need to find this radius right here so let’s call this distance right
here in x so that would be the distance from the center right there because
we’re revolving around the y-axis here so yeah this is going to be an x
out to the rectangle right there and so you know if you if you uncoil this

00:04
because it’s been revolved right but if you uncoil it then it looks like a
rectangle but it has some thickness to it
so i’ll put the thickness right here so that thickness here we can
put in blue again and the thickness is a little tiny
change in x so let’s call this here this distance right here delta x
and this distance right here so right so that’s going to be the circumference
when you unroll it right here that maybe let’s make that a little bit more
straighter straight um and anyways in any case what is the circumference going
to be it’s going to be 2 pi times the radius right so there’s the
you know that’s just how you calculate the radius right and so
how do you find the volume of this right so the height here
is given by so the height of the rectangle is given by the function
and so i so how do we find the volume so a little tiny volume element because

00:05
this is tiny here right this is not tiny necessarily and
this is not tiny they depend upon x so does the delta x but we want the
delta x to be smaller and smaller as we integrate so anyways the volume
will be you know because this is a square you know not square but a
rectangular solid right so it’s uh 2 pi x times f of x
so with times height times deepness i’ll say delta x here
so in any case we can uh so this would be like an estimate for the tiny little
volume and we’re going to add up all the little tiny volumes and we’re going to
do that by passing a limit here so the volume will be given by the
integral and we’re going to integrate from 0 to 1
and we’re going to have a 2 pi x f of x and then this will be dx here and so
this right here will give us the volume and we can find this volume just simply

00:06
by integrating this out here so let’s take out the 2 pi so we’ll say 2 pi
integral from 0 to 1 of an x and then times our f of x which is
x minus x to the third and so that looks really good
and so yeah we just need to multiply the x out and then integrate
so we’re going to get 2 pi and we’re going to integrate 0 to 1 of
minus x to the 4th plus x squared dx and then if we integrate this out here
we’re going to get 2 pi times minus 1 5 plus 1 3 and we’ll evaluate that 0 to 1
and so for that number right there we’re going to get 4 pi over 15.
and so that right there is the volume 4 pi over 15.

00:07
and so you know when we revolve this region around the y-axis we get a solid
the volume of that solid is 4 pi over 15. excellent so there’s our first example
so let’s see how to write this up into a theorem or into a into a method and to
get this idea more formalized so let’s see that now
so here’s the theorem right here but let me erase this real quick so here we go
um yeah so we’re going to take a continuous function
non-negative on in our last example we used f of x was equal to x minus x to
the third and we were on a bounded region zero to one so we can be
uh you know from any a to b and we’re going to take r to be the
region under the graph on the interval and so the volume will be given by
and so we’re going to be adding up a whole bunch of volumes so in the

00:08
previous screen we said what delta v is a little small a volume right here so
think about this as is the circumference times the height times the depth
and and so we’re just going to integrate this out we’re going to take the limit
of the riemann sum right here so um you know when we
um write this out you know we could write it out something like this right
here we could say equals a to b and i’ll say 2 pi x
and then i’ll put brackets here f x and then i’ll say d of x here
and this part right here you want to think about this right here
is the circumference and you want to think about x is the radius of your shell
and you want to think about f x here as being the height

00:09
and then you want to think about the dx here as being the thickness and so
this is a nice easy way to remember the shell method here
so when we have a shell right here let’s say we’re coming down here
and we have a shell [Music] and let’s let’s give it a little thickness if we can
and so it’s going to come back around and then it’s going to have a little
thickness to it so let’s call that let’s come in here with a little thickness
and so yeah it’s going to come like this and then around here like this
and it’s going to have a little thickness in here
and so that thickness right there that’s our d that’s our dx
and the height is given by the f of x and the circumference right here well we
need to know the x which is the radius right here and so yeah that’s going to

00:10
give us the shell method right there um you know if we unroll this right here
i call it unrolling or unwinding [Music]
yeah we get the height f of x we get the circumference 2 pi x
and we get the thickness right here which is the dx
all right so circumference 2 pi x here if we unroll that we get the 2 by x there
yeah so 2 pi x and so you know when we when we do the shell method we want to
kind of identify each of these things here
and not just use this formula here and the reason why is because
this is stated in terms of revolving around a y-axis but we’ll do some
examples where we revolve around not just the y axis or
the line x equals zero but how about some other vertical lines we may want to
revolve around them moreover we’re actually going to turn this on its side

00:11
and do it um you know instead of integrating with respect to x we can
integrate with respect to y and in fact we can
uh revolve around any horizontal axis so
we’ll we’ll do all that here in a second also
so yeah let’s look at an example now so here we go let’s see another example
all right so now we’re going to take this line here y equals x
and y equals x squared and we’re going to revolve this around the
about the y axis there we go so here we go let’s draw a sketch right here
and so we’re really just in the first quadrant here
um and so let’s see here we got y equals x coming right through here
and we got y equals x squared coming right through here
and they’re going to intersect right here at 1 and

00:12
we’re going to revolve this region right here around the y-axis so [Music]
and let’s shade the region that were that we are revolving
so this hits right here through the origin and so let’s just shade this in here
really quick and then we’ll draw a representative all right
and now notice our representative here is going parallel
to the axis of revolution as compared to the washer method
but in any case we’ll do another episode where we compare those methods um you
know in particular right so right now we’re
just learning the shell method and we’re going around a vertical axis which
happens to be the y-axis so we’re going to revolve and we’re going to get a
solid and so what i want to do is i want to think about how this representative
would actually become a shell and so when i revolve that i’m going to
get a shell and so i’ll put that shell down here

00:13
and it’s going to have some kind of thickness to it and
so we’ll come down here and draw another one
and it’ll have some kind of thickness to it and so what will the height here be
the height here of a shell right here won’t be um
just f of x anymore because we’re trapped between two
so the height here is going to be given by
uh this height right here and then take away that little pi part in there that’s
not there so the height will be x minus the x squared here so again to
get the part in blue we just do the x all the way up
and then minus that little part right in there
now the circumference right here so we’re going to need to measure out all
the way to an x all right so the circumference will just be
the uh 2 pi x so here we go here here we can go we can find the volume so we’re

00:14
going to integrate 0 to 1 and our circumference is going to be 2 pi x
and our function here is going to be i’ll just write 2 pi x
and it’s going to be x minus x squared and then dx
and so this right here will be the volume right here and so this right here
will be a zero to one we can take out the two pi
and then just say x squared minus x to the third and then
we need to integrate this out right here we’re going to get 2 pi x to the third
over 3 minus x to the fourth over 4 and then let’s just evaluate this right
here zero to one all right it’s looking good
and in any case when we come in here with the one
we’re going to get one third minus one fourth
and then when we work on this all the way out we’re going to get pi over six
when we reduce that all right and so that that’s going to be the volume

00:15
when when you generate that solid right there
all right so there we go there’s our second example there
and let’s look at another example now so this time we’re going to take this
region right here um 2 x squared minus x to the third
and so let’s just draw a sketch right over here
so we’re going to come in here it’s going to come in here like this and go
up and then come back down and it comes right here about a 1
and about a 2 right here and we’re going to revolve rotating about the y axis
again so let’s come in here and revolve and let’s shade our region and
let’s just think about how [Music] you know if you don’t know how to sketch
this graph just off the top of your head which you know probably not but

00:16
in any case just do it you know maybe a derivative test or something like that
but anyways there’s our region right there that we get and
now when we draw a representative let’s say our representative is about right
here and now when we revolve that we’re going to get our cylinder
so we’re going to we’re going to come over here and sh oops we’re going to
come over here and choose an x choose an x right here
and then the delta x little tiny distance there
and then our height is from the uh from the x-axis up here and the function is
um 2x squared minus x to the third all right so once we identify those
things there we’re good to go so here we go the volume is
we’re going to be integrating from 0 to 2
and so the circumference is going to be 2 pi and then we got an x here we’re

00:17
going all the way from our axis of rotation out to
a chosen representative so 2 pi x and then times our
height here which is 2x squared minus x to the third
and then times our width of the representative was just dx here okay
all right so there we go so now we can pull out the two pi and go
from zero to two and let’s look at two x to the third minus x to the fourth here
dx all right and so we can skip a couple steps here this is 16 pi over 5 right
here so right you can calculate that right this
is this is usually a calculus two subjects right so you should be able to
to calculate that but double check that you can and you should get 16 pi
over five there all right so there’s our third example
there let’s look at something a little bit different now

00:18
so now let’s go about the line x equals two so here we’re revolving around the y
axis which is x equals zero so now let’s revolve around something a
little bit different and see how this all changes here all right so here we go
let’s do another call to action here there we go
very good all right so um example four here
so we’re looking at x to the third plus an x plus a one
and so what does that look like here on this region here so here we go with a
sketch and so we’re going to go around x equals 2
so we’re going to need lots of space over here
so let’s come in here uh something about like this
and then we’re chopped off at x equals one so we’ll put a one there
and then what’s the height here let’s chop it off here uh the height

00:19
here when x is one you get three and then so let’s just come across here
like that so y equals one right here when x is zero
and so we’re going to get this region right here so um actually
let’s try that again that didn’t look good so this is going to come through here
like this and this is going to be at 1 and we’re we’re right here y equals 1
and x equals one all right so there’s our region there
yeah so the height is three and this is a one but this is our region here
all right very good so there’s our region there and now what are we

00:20
revolving around x equals 2 and so let’s put that here
so let’s sketch this over here let’s say there’s a 2 right there
and now let’s go out here and say here’s our axis of revolution
at x equals two here so let’s just put it here x equals two
all right very good so now let’s draw a representative
let’s draw a representative so we’re revolving around a vertical axis here and
so we’re using the shell method so we’re going to choose a parallel to the axis
of revolution and there’s our representative right
there and so we need to think about how we’re getting a cylindrical shell out of
all this right here so we’re going to revolve around this axis right here
so what is going to be the circumference remember the circumference uh
you need 2 pi times the radius it’s measured from the center of the

00:21
axis of revolution so we’re going to be measuring the radius from right here how
do we get this distance right here right so this distance right here is going to
be let’s just put it right here so that distance right there is going to be 2
and then take away an x right in here so i’m going to say this is 2 minus an x
and then i’m going to get our little tiny width right in here which is delta x
i’ll just put a little tiny double arrow right here delta x
and so what’s our height going to be so we know how to get to this point
right here that’s just the function let’s call it uh x to the third
plus x plus one let’s just call it f and so that’s going to be
f of x but we’re taking away this part right here which is one
so this height right here is f of x minus one
okay so i think we’re ready to find the volume here we go volume is
and we’re going to be integrating from zero to one

00:22
um and so we need the circumference which is going to be 2 pi times the
radius which we’re finding to be 2 minus x and then times the height here
which is x to the third plus x plus one and then i’m going to just say minus one
just so that we can see how we got that you know so plus one
minus a one it’s a plus one because that’s coming from the function
and then it’s a minus one because in order to get this height here it’s the
total height f of x and then take away this one here and then the left over
part is the representative height right there
all right so let’s pull the 2 pi out and integrate 0 to 1
and when we expand this out we’re going to get minus x to the fourth
plus two x to the third minus x squared and then a plus two x dx
and so we just need to integrate this right here

00:23
just the power rule piece by piece work it out and we’re going to get
dot dot dot we’re going to get 29 pi over 15. 29 pi over 15
29 pi over 15 almost two pi’s all right so there’s an example there
we’re revolving around a vertical axis that’s not necessarily
the y-axis right there so you got to pay attention really to how you find your
height and how you find your radius right there
so yeah let’s look at another example but first let’s look to see how we’re
going to switch it up and integrate with respect to y
so that right there is just a um you know integrating with respect to y so here
it’s going to be the exact same thing it’s going to be limit of a sum
and we’re going to have our circumference this part right here is
the circumference this part right here is the height

00:24
and this part right here is the depth and so we’re finding volume by by
multiplying those three things together and then adding up all the little tiny
volumes and then passing a limit so making the uh little tiny estimates uh
get smaller and smaller and smaller so that you would get the exact value from
right here notice here that we’re evaluating along the y-axis right so we
still need a continuous non-negative function we still need an interval but
this is along the y-axis and r is still the region bounded by the graph
so pretty much here everything’s the same so let’s do some examples now
yeah let’s do some examples so example five here let’s look at using
the shell method here but this time we’re going to revolve around the x-axis
right here so let’s look at the sketch of this region right here

00:25
let’s put it right here and so we got this y equals square root of x
and let me make it a little bit more straight straight
and we’re revolving around the x-axis so let’s make this a little bit longer
and go around the x-axis and you know we got the square root of x and
the line x equals four so let’s put x equals four about right
and so now we’re looking at this region right here
and we’re revolving around the x-axis right here and we’re going to
take our representative here parallel to the axis of rotation and so

00:26
we’re going to choose a representative here and now instead of using a
x we’re going to use a y so this is uh y equals uh f a function of x
so let’s change this into x equals y squared here uh think about this as
part of the parabola sideways parabola x equals y
squared and we’re just looking at the top part here so this will be our g of y
here um assuming y is positive right there so so that’s how we’ll
calculate this uh width right here which in our previous examples was our height
but now it’s our width but any case we have a little delta x here
i guess we’ll make that in blue so that’s our delta sorry delta y here
that’s our thickness and now so we want to measure the radius
and we want to get the height okay and so actually this is a two um when

00:27
x is four yeah x is four then this height here is a two um
you know think of this as square root of x right so plug in the x is 4.
so anyways we’re going to be integrating 0 to 2
all right so i think we’re ready to find the volume here so the volume is going
to be 0 to 2 and we need to find the circumference so we’re measuring from the
axis of rotation right here and so that’s just going to be a
y height right there and so this will be 2 pi y and then times the width of the
representative right there and the representative is going to be 4
take away that little part in there which is going to be given by this right
here so this would be 4 minus y squared and then d y
so this right here will give us the volume right here we’re thinking about

00:28
this is the circumference and this is the
not the height anymore we’ll think about this as the width
so when we revolve this around the x-axis what we’re getting now is a
cylindrical shell this way and so this has a little bit of thickness to it
the thickness is given by the um blue blue part there
so we’ll just have this little thickness in here
and we can find the volume of this object here by uh calculating
and maybe unrolling it if you want and we can find the
um so right so if we unroll it now we’re going this way and
right and so we’re going to have this width right here is going to be
given by the 4 minus y squared and this will be 2 pi times y
so y is the height right here you just choose a y

00:29
and when we were integrating along the x axis we chose an x right we choosing an
x for the representative right there a y sorry so 2 pi y
and then okay so let’s integrate this out so um you know let’s just pull the two
pi out and go from zero to two then i say two pi right here two um zero to two
and pull out the two pi and so we’re just going to get here
4y minus y to the third d y and yeah if we integrate this out we’re
just going to get here 8 pi that’s what it reduces to after you finish
integrating that out we’re just going to get 8 pi so we’re going to revolve this
around the x axis here and we’re going to get 8 pi for that volume there so
this is the volume here because it has a little thickness to it which is the

00:30
delta y all right so yeah we got this representative
revolving it we’re generating a region right there and let’s look at
one more example last example here we go now this one is is uh fun because we’re
going to revolve around a um horizontal line but it’s not going to be the x-axis
so let’s look at this right here let’s try to sketch a good sketch here because
we’re revolving around [Music] y equals two so we need to get a good sketch here
and that’s not going to be good here we go let’s try again so

00:31
we have here y equals x squared i don’t want to make it too uh just i want to
like uh come out here with it and we’re stopping right here at x equals one
and so let’s just block that off right here at x equals one
and so we’re looking at this shaded region right here
this region right here is the region that we’re going to revolve
and we’re going to revolve it around let’s see if i save myself enough room
around the so this is 1 1 squared is 1 and let’s come up here and say this is
about a 2 right here so this is y equals 2 right here
and we’re going to revolve around y equals 2 right here
so let’s revolve around this line so we’re going to get a three dimensional
solid right there and revolve around y equals two and so we’re going to take our

00:32
representative horizontally and so we need to think about in terms of radius
and the width and that kind of thing right here so here we go so the volume
is going to be what are we going to integrate on the y axis here 0 to 1
and what’s going to be the radius so the radius here we’re going to have a
remember we measure the radius from the axis of revolution
right just like if you have any circle the radius is measured from the center
of the circle and when we’re revolving around the y equals two the
axis of rotation that that’s going to generate the center so you know this is
say this right here the height here is y so this distance right here
is going to be 2 minus y so two all of it and then take away that

00:33
little part right in there so the radius here will be uh two pi um
and then two minus y and then how are we going to get the width here
so the width is a full 1 take away this part right in here
and remember this is the line this is the
y equals x squared so if we solve it we can say y equals square root of x
so that we can get a function of y right here and so we’re going to say
square root of y here and then d y to get the to get the thickness all right so
yeah think about that as a full one and then take away this little part right
here all right and so let’s just pull the two pi out
and then we’re going to go zero to one and then we’re going to expand all this
out here and we’re going to say 2 minus y minus 2 y to the one half

00:34
and then plus y to the three halves and then integrate that with respect to y
and then after a little bit of work here
we’re going to simplify simplify that to 17 pi over 15
as the volume of the solid that’s generated it’s going to have a it’s
going to be hollowed out it’s going to have some kind of missing part of it
because the region is not flush with the axis of rotation but we’ll get a solid
and there’s the volume of that solid right there
excellent so if you enjoyed this video and you look forward to more videos then
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About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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