this is my tradition whenever I teach calculus one (volume of solid of revolution, disc method, washer method, and shell method), my very last lecture example is always on
how to find the volume of a donut aka volume of a torus and i also buy my students Krispy Kreme
donuts just in case if they never had that before so you can imagine we'll all be eating donuts and do
calculus sounds a lot of fun right yes but after an hour we all get sugar crashed but anyway today i
will show you guys three ways on how to find the volume of a donut (by using the washer method, shell method and the Pappus Theorem for volume) but to make this more
interesting of course we'll find the volume of an actual Krispy Kreme doughnut so have a look right
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firstly this is how you can draw a donut no not yet i know this looks like a potato but
if you do this and that hopefully looks more like a donut now
and then we can see that the cross section here is a circle and the idea is that we have to take
the circle and rotate about the y axis this is how can generate a doughnut aka a torus
all right and then we see that the measurement from here to here is about three centimeters and
then from here to here it's about 9.5 centimeters all right so now let's go ahead and put this down
onto the x and y plane so that we can use an equation later so we have a circle here first
and then rotate this above the y axis so i would just put down like a mirror image here and then
i'll just connect them like this like that so this is how the donut looks like from the side
all right and now let's put on some numbers on the x-axis
we see that from here to here is 9.5 cut into half that means from here to here it will be 4.75
and then from here to here is three so that means we subtract three this right here will be 1.75
and the radius is 1.5 so just at 1.5 we get 3.25
so now we need an equation of this blue circle it's not bad because we can just use the standard
form here's the center 3.25 comma zero so we have x minus 3.25 and then we have to square that and
we add it with y squared and that's equal to the radius which is 1.5 squared good okay so
how exactly do we get the volume of a donato like this and this is called the solid of revolution
the first method is we can use the so-called washer method it's pretty much the disk method
for the outer piece minus the this mag the disk method for the inner piece
so i'm going to give you guys two pictures right so the first picture it will look like this we
need to know the equation on the side right here the outer piece so that when we do the rotation
we can get the outer piece like so so this will be the whole thing yeah but we do have
a hole in the middle because it's a donut so we will have to subtract so here is the
subtraction subtract the volume of the inner piece so the inner piece will be this part right this
part technically it should be right here so i will take this and then mirror it image that and then
that's like the hole in the middle so that's the idea and of course all this part right here
they are meant to be perfect circles so now this part is the x equation from here but that's the
positive version you'll see let's go ahead and just solve x from here and we will end up with
two equations because when we take the square roots we have to do the plus or minus
firstly let's put this to the left so we will have 1.5 squared and a minus
y squared and then of course we have to take the square root now so we take the square root and
then don't forget that we have the plus or minus and then move this number to the right hand side
so 3.25 here so we do have two equations this one is the one with the positive so i will write down
and this one's the one with minus
so let's take a look here first we are using the disk method for this and to do so
the falling of a disk again just like this very thin cylinder we need to know the radius from here
to here and that's exactly the horizontal distance namely just x and then the vertical distance is
the height with the thickness and because this is just a small change in terms of y so that
will be the y and for this i will just put this down right here for you guys a volume of a disc
or a very thin cylinder it's equal to pi times the radius squared times the thickness what the height
so that's pretty much the idea so for this one we are going to get pi times the
r which is x which is just this part and we have to use this right here because later
on we'll be in the y world we have the y for the thickness so i'm just going to write down
this is the radius we will have to square that and then we multiply by the thickness which is d
y but this right here is just the volume of the red disk we want the whole thing
therefore we will have to integrate this and because we're in the y world we have to
pay attention to the y values the starting and then the end well let's look back to here again
the radius of the blue circle is 1.5 and then if you look down here
the lowest y value is negative 1.5 and this part is just so that i can make the picture look as 3d
so you look at this very button it's negative 1.5 and then this right here to the circle is 1.5
so we will just have to integrate this from negative 1.5 to positive 1.5 this
right here is the volume of the outer piece assuming that you do not have like an actual
donut with a hole you have like a boston cream that kind of thing all right but we do have the
center that's missing so we will have to subtract this so minus pretty much the same thing integral
and then we have pretty much the same thing but just this right here change that to a minus
so this right here will be the setup to get the volume of the donut and usually
for this kind of questions i would just focus on the setup because usually a lot
of students will mess on a setup if you want the students to also do the integrals by hand
it doesn't make sense anymore because if the setup was wrong right so once we set this up
let's just use the graphing calculator and if you work that out you will get approximately 144.343
centimeters cube so that's how big a crispy cream tonight is
i don't know if i did this right wrong because usually this is pretty hot right so let's go
ahead and try to use the shell method so for the shell method the idea is the following we are
going to make a vertical cut right here and then we are going to wrap it around about the y-axis
so we are going to get this right here as well and each wrap has a volume it's like a really thin
paper that kind of thing and the idea here is that we will have to know the folding of that
thin paper because we see that it goes around like so so we would need to know
from here to here and that's again the radius so i'll just put down x for now because the
radius is horizontal so that's why it's x but the idea is that it's 2 pi times the radius
that's going to give you the whole width of the paper and then we will also have to multiply by
the height right so multiplied by the height of the paper because it's going to be like a
rectangle but this is just the area of a paper rectangular paper we also need to multiply by
the thickness so that we can get the volume so last thing that we do is multiply by thickness
so this is the volume of each shell which is again the volume of each like a vertical sheet
of rectangular paper all right so now as we can see the r is the x that's good what's the height
well the height from here to here is the vertical distance which is y and then from here to here
guess what it's technically negative y because it's on the bottom but we're talking about volume
so which is 3d as possible as well so the whole thing is 2y all right and what's the thickness
though well look at how we draw it right here this time unlike earlier the change is in terms of x
so this little thing right here is dx so this is the advantage of using the shell method over the
disk method was that the washing method earlier we don't have to do any subtraction
but anyway let's see this right here again we are in the
x world okay and then let's just continue we have the 2 pi that's good and then the r is just x
and x is a lot in the x world so that's perfect i'm just going to put down x however y is not
allowed in the x world so from here we will have to solve for y so let's go ahead and do that
pull this to the other side and then just take the square root we see that y
is going to be plus or minus square root of 1.5 squared minus parenthesis x minus 3.25
squared like that but again in our case here the h is technically
2 y so we are just going to multiply this by 2 and then the y is just that
so this is going to give us the volume of each shell and then we just have to integrate this
and now we have to worry about because we're in the x world
so look at the x value it starts at 1.75 up to 4.75
so this right here will be the setup for using the shell method and again we'll just focus on
the setup right so once we have this setup let's just go ahead and use a calculator to solve it
we get about one point we get about one you know the same answer 144.343
centimeter cube of course we did this right earlier as well
so that is what we have now i want to show you guys the best way to find the volume of a donut
this right here is called the purpose serum and i'm going to do a proof for you guys later on
this is what it says volume is equal to area times distance
too simple but i'm going to explain the distance in a little bit
let me give you guys intuition first well how do we find the volume of a cylinder
the volume of the cylinder is pi r squared times h isn't it what's pi r squared well the
cross section here is a circle this is just pi r squared good and then why do we multiply by h
the h is the height but you can see that it's just pretty much the area here multiplied by
the distance that you have to go all the way from bottom to the top isn't it yes so that's the idea
and technically for the d right here we have to talk about the so-called centroid but since we
are talking about circles the centroid is just literally right in the middle that's all so i
will tell you the distance here is technically the distance traveled from the centroid so now
let me draw a donut for you guys again like so well we have our cross section which is just
a circle and the area for that is just pi r squared good now what's the distance that i travel
it will be like this right like this like this way like this way yeah so it's also going to be
traveling a circular distance so it's going to travel like this right here and again we have to
measure it from the center aka the centroid so what's this distancing bludo well that would be
using this center right here as the center and then you have to look at the radius here because
this is the circumference for that blue region for the blue circular pathway i'm going to call
this capital r and then the distance that we need here it will just be 2 pi r for the circumference
so ladies and gentlemen thanks to papa's theorem the following of a doughnut or torus
this is just equal to pi r squared let me just put this down right here
and then we multiply by two pi capital r of course we can simplify this a little bit this is just 2
and then we have pi squared and then r squared and then capital r this is the volume of a doughnut
so now let's look back here okay what's the radius of the blue circle well from here to
here is just 1.5 so ladies and gentlemen volume is going to be equal to let me just put this down
pi times 1.5 squared all right now here's the harder part what will be the capital r
again we use this as a center and then we would just have to make sure that it traveled like so
and then that's the distance right the radius for that okay so what's the distance from here to here
remember the picture that we did earlier 4.75 minus 1.3.25 this right here is 3.25
so anyway we just have to multiply by 2 pi times 3.25 yes we will get approximately 144.343
centimeters cubed this is definitely the best way to find the volume of this donut
how do you get stick which method did you like the most and let me also know that
how you did in your account one class this is the summer time now so hopefully everybody will
have a good summer and if you want to make your summer productive go ahead and check out polyjets