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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

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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

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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

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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

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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

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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

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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