finding the volume of a Krispy Kreme donut by using calculus (washer method vs shell method)
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- čas přidán 27. 06. 2024
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We will find the volume of a Krispy Kreme donut by using calculus! I always show my students how to find the volume of a donut (aka the volume of a torus) during my last calculus 1 lecture. I would also buy my students Krispy Kreme donuts! Here we will find the volume of a Krispy Kreme donut with the washer method, the shell method, and the Pappus Theorem for volume. This example is good for calculus 1 students and AP calculus AB students. Some students might do the volume of solid of revolution in calculus 2.
For more examples on volumes of solids of revolution by using the washer method vs shell method, please see 👉 • Disc/Washer Method vs....
0:00 intro
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1:51 volume of a donut by using the washer method
8:55 volume of a donut by using the shell method
12:53 volume of a donut by using the Pappus Theorem
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#calculus #apcalculus #krispykreme #math #gcse #mathteacher #calculusteacher
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Nice vid! I had fun verifying that the 3rd method does give the same answer as the 1st method for the elliptical torus [🍩] consisting of revolving an ellipse with horizontal axis a, vertical axis b, and whose left-most point is R units to the right of the origin! V=2π²abR :)
@@MathNerd1729 I believe a circle is revolving, not an ellipse
@@Jasomniac Well, a circle is a special kind of ellipse where a = b. But you're probably right that it technically may not be a torus! I meant that I liked verifying it works when you revolve any ellipse with horizontal and vertical axes around the y-axis in a similar fashion regardless of whether or not it's a circle! :)
@@MathNerd1729 sorry, I read your comment wrong, what you said is well phrased. And btw, I don't know if "general torus" is the right phrasing, but if we're gonna be very specific, I guess you could have said elliptical torus.
@Math Nerd 1729 but when you say it works with elliptical torus, do you switch the "r^2" in the last method for a•b ?
How a chemist would solve it: Put in beaker and see how much the water level rises.
that would be the best method if you had a real donut (if it was waterproof), because of small irregularities on the surface of the donut that are hard to calculate
Archimedes: "Eureka!"
@@herman7550 Except, once waterproofed, it would probably float. You could still do it, though, if you used a very low-volume implement to force it to submerge.
Fred
Chemists are water addicts
@@ffggddss It'd be trivial to correct for the volume of the implement used
using tripple integral to find the volume reduces to these equations depending on order of integration
Indeed. I must do calc over this summer.
Its too late lol
Gosh same
Bet you say that every summer
Im not even in calc yet, im still in algebra. But my curiosity always push me beyond, just wandering here :))
A donut is like a cylinder which is curved into a circle of radius R....hence height of cylinder is 2πR...and if the radius of the cylinder is r then area of cross section will be πr²...hence volume of the donut will be the volume of the cylinder = (πr²)*(2πR) = 2π²r²R
That's actually how I solved it😅
Edit: after watching the full video I realised that's the pappus method😅
Yup! 😃
If you morph a cylinder into a torus like that though, parts of it are going to be compressed and other parts are going to be stretched. Apparently if you choose your radius of rotation to be the centroid of the circle, the compression and stretching cancel out. It is noteworthy that you have to make sure you use the centroid though, and if you're rotating something other than a circle around a central axis, it may be difficult to figure out where the centroid is.
@@IGDZILLA That is true, but in the limit as thickness of the wedge(distance between centers of circular bases) approaches zero, the difference between outer and inner "heights" will also approach zero and the wedge will approach a disk shape
@@hiimgood That reasoning is tricky though. That would apply to any cross-section.
@@chitlitlah I noticed it and tried to find the average of the compressed and the stretched side and realized that it IS the centroid
If you're doing it by disc or shell, you can simplify the integrals by noting the symmetry of the problem.
im in high school and ur videos brought me into calculus
and for this summer i planned to take calculus classes
(overall i really enjoy watching ur videos and u seem like a very kind guy urself)
I watch all your videos, you are one of the best mathmatians in the world
Maybe not the best mathematician, but definitely the most entertaining 😊 (He's definitely very educated when it comes to math though)
Thank you. I am definitely not the best mathematician but I thrive to become a good math teacher/CZcamsr 😃
I just dip in water just like Archimedes did to find the volume.............
You inspire me to learn calculus. Keep up your great videos!
I've just had my last lesson of mathematics seminar this year on high school. It was about calculating volume of shapes with functions and integrals and a simple formula.
Very nice presentation! Thoroughly enjoyable. Some related thoughts it brings to mind:
1) For the benefit of other viewers, in your example of Pappus' Theorem - for a torus, r is called the "minor radius," and R is called the "major radius."
2) We were taught Pappus' Theorem (P.T.) in my 7th grade general math class. It was a very impressive example of mathematical beauty, as well as being useful for many otherwise very hard problems. Then, years later in calculus class, one of our exercises was to prove P.T. - the general (3D) case.
3) And just BTW, it also works for finding surface areas. You might consider also finding the *area* of the Krispy Kreme this way ;-)
4) P.T. generalizes to n dimensions, for n ≥ 2. You can find the area of a circular annulus by revolving a line segment around a point collinear with it, and outside of it. You could find the 4-capacity of a 4D solid of revolution, given the volume of the 3D solid being revolved, etc.
4a) I once used it to find the capacity of an n-ball, using an insight that a certain relation between a 2-sphere and a circular disk, might generalize to other dimensions.
It involved a torus whose major and minor radii are equal (a zero-hole torus); and I was able to verify that generalization formally.
I hope, one of these days, to make a YT video about that...
Fred
Thank you, Fred!
Thank you for the great videos! I saw your channel for the 1st time 7 mo. ago on the derivative on x^x, and I didn't even know what calculus was yet lol. After watching many of your videos, I now know lots of calculus 2 thanks to you :D
Happy to help! Thank you.
I only recently discovered your amazing channel. I'm a teacher and I would have loved to have a resource like this when I was a student. Subscribed!
For a complete newbie in calculus, this makes so much sense, especially the Pappus Theorem! I accidentally saw it on my calculus book while cleaning my shelves, but your explanation made the theorem more exciting and sensible. Awesome content, as always
I absolutely prefer the Pappus Theorem, because it is the most intuitive to me (and it avoids integrals... I haven't done integrals in 20 years).
Hey man love the vids!!!!! I'm Spanish (studying in Spain) and I got a 93/100!! Really appreciated your help with the integral videos😄😄😄
What the heck, I never had pappus theorem taught in Calc 1 THAT WOULD HAVE BEEN NICE!
You are a Great Teacher!
Omg I was thinking for the whole video that the Pappus-Guldinus theorems would be so much easier, and then you did it; well done.
Wish I had a math professor like him :(
Its funny because I was thinking of how to solve this withouth calculus, and I thought of using the circumference of the big donut times the area of the cross section. Turns out it was the circumference of halfway through the big donut, but it was still cool that I almost figured it out on my own.
wow, never knew there is so much to learn from a donut
You are a great teacher my man
Why doesn’t this have 4 million views? It’s literally the perfect video for the algorithm
I actually used this approach years ago to calculate the volume of a vase in my house. Afterward I filled the vase with water and I was only off by about 1% in accuracy. Fun stuff.
Love this idea!
Good video!
It would be interesting to see a solution with Fubini and Tonelli theorems
And maybe with the change of variables to cylindrical coordinates
According to my mentor and professors, I am very good at calculus. I feel that way too but not as much. The thing is, bprp makes everything seem so SIMPLE!
Hi man, I just want to let you know that you helped me with gaining more motivation for math. I am from the Netherlands btw. Thank you sir
I am happy to hear that my content has helped you 😃
⭐ What a brilliant video! ⭐
Practical integrals at its best!
I actually begin questioning into the region of integrals and limits when I questioned the volume of a cylinder in class IX. I wondered how one could find the exact volume when the circle area had infinitely small thickness and was multiplied by the height. Now in Class XI, I see how integrals can be used. Epic!
😄
I got it! But by using none of these ways, sort of (I actually did do the Pappus theorem, but solved for it first and the best part is I didn’t know it was a thing when I solved it). I basically did the washer method, except I did it the easy way. Which is by turning everything into rectangles, which ends up turning everything into cylinders.
I solved for it generally, and then plugged in the numbers.
The result was
2m(pi^2)(r^2)
Where m is the distance from the center of the torus to the center of the solid circular part and r is the radius of solid circular part.
And as expected, the approximate result was 144.343 cm^3.
We need more teachers like him
Well,I like the second way to solve the volume of a donut.It’s pretty clear and easy way to learn😀
Wow! Just amazing!
Yooo congrats bprp on the sponsor!
Thanks.
I'm deeply impressed that in the US, you've got rulers divided in cm!
Now I will make my summer productive by choosing the donut with highest volume 😋
cool video man!
This is a very creative idea
Thanks PROF UR the best)
i could solve it by the pappus theorem before watching the video:)
if you had a function f for the circle, you could do pi times the integral of f^2 to also get the volume of the donut
Yes, i thought, he would mention this way, it’s the way I solved this problem at school.
how have i not heard of you sooner...this may honestly be better than khan academy
I was thinking Pappus theorem, I also had the top marks in all 3 OAC (grade 13) maths: Calculus, Algebra, and Finite. Ended up taking a few more Finite and Probability as electives in my undergrad. Grad school did offer a course in Numbers but it was Pentateuchal studies.
I like your videos... And earned my subscription ❤️
In one Homework in mathematics for Physicists, we had to also do that. I remember having brought this up to the table with Pappus theorem, but we couldnt use it, because it was an higher analysis course and we should have done in the 1st presented way.
It was challenging but also fun, when you get your results correct :D
By the way, i didnt know, that this theorem was called "Pappus Theorem"...
lol. That end ist good "Just make it easy i bored you"
great.I like the second method
I think this would work too,
find circumference of outer and inner circle, find average, and multiply by a cross section of the donut.
Finally some real word applications for maths
That use either shell or washer method. We actually didn’t learn that in calc 1 rather it was in calc 2 cause calc 1 was rushed over a summer interim session, but nonetheless it seems odd that such a simple concept requires calculus. Interesting video, the only thing I can comment on at the start is that you will have to do two times the integral of the positive half of the circle offset by some volume.
The shell method resonates best with me.
The Pappus Theorem feels like a trap. Not that I think Pappus was lying; it's just that it feels like the sort of thing a person could misapply if they're not careful.
A while back I came across your world record video and always wanted to learn calculus. I think I will succeed this summer with the help of your videos! Thank you!
p.s I had to laugh really hard during the wr video @4:55:55 ''Integration by parts!''
Hello, thank you very much for this amazing video! I was also trying to calculate the volume of an elliptic torus shape revolved around the y axis by using the Washer method. Would this formula be applicable to ellipses as well?
Can you explain when and why you pull out dy/dx? Never really understood it
You can also measure volume by dunking donut into a cup and you can see the difference between before and after
Can you answer me this, in what country and in what school or college do you teach math? Because it seams like your students are having a lot of fun with you
Probably some differences in naming. When I studied we called the third method "the 2nd Guldin's theorem".
The 1st Guldin's theorem finds the square of the rotated curve.
I got an A in my calc I class - 45 years ago!
48 for me. (Had to pull out a calculator to figure that).
Donut is a cylinder with height 2πr
Volume of cylinder = πR²(2πr)
2π²R²r.
Long ago i calculated it in my mind
π² term facinated me!!
explaining to your audience why you have to use the centroid would be useful for this Pappas method. reason being that the inner radius is shorter than the outer radius makes it seem as though modelling it as a straight cylinder could be inaccurate. using the centroid sort of "averages" the over estimation of the inner radius and the under estimation of the outer radius. something like the area of a trapezoid where you average to two unequal sides. same idea in an abstract sense. it took me a little thinking to agree with this pappas method. so a better understanding, say for example if the cross section was not a circle, but say, two half circles with different radii forming a funky donut (torus). then you would have to integrate the cross sectional shape to find the centroid, then use the coords of the centroid to apply the Pappas method.
nice video, I liked how you used three methods.
I was calculating this with the first method, albeit I changed the x and y-axis. I also tried to calculate its surface, but I was not able to find the proper solution.
Save this man
Finally some actual real world use for math!
I didn't learn volume of solids by rotation until calc 2. Also, I never learned about Pappus Theorem although it showed up in my engineering statics book. I took calc 1 two years ago and got an A.
Now N A S A will have easy time calculating the orbit of satellite circling around the planet. Thanks to Pappus theorem. Thanks a lot for your video. Super informative 👍👍👍👍👍
Could you please do a video where you calculate an egg's volume?
The teacher is very smart enough. I am very happy to learn from him to solve this problem. I hope the teacher can get NASA recommendation to help astronauts to solve Mars, Moon or the universe engineering calculations problems.
Exact value: 14.625pi^2
Don't you need to do trig substitution to evaluate the integral? Which usually isn't taught till Calc 2.
Can you make a video on the volume of a Croissant?
Me: *blending the krispy kreme fine and using a measuring cup*
Engineers: Welcome to the approximation appreciation group
what if we view the doughnut from the top (y axis pointing towards our eyes) and then convert everything to r and θ, and integrate from θ = 0 to 2π
I've always wondered how loud they were.
哈哈哈好眼熟 指考生寫積分的時候有寫到🤓
would have been nice to add in the cm after the numbers, so we can really get the cm3 :)
my calc teacher did this for us too
Will you still upload over the summer?
Yes
As an always hungry mathematician
That was so delicious 👏
The integral of (arctan(1/x))/x
Hi! Can you integrate -(cosx)^3/(sinx)^2 for me? I've only learned sin, cos, tan, arctan and arcsin yet :)
My intuition of what the answer would be was something like, "take a cylindrical bar of dough of that radius, bend it to circular shape where both ends meet without distorting the radius of the cylinder. The volume of that cylinder will equal the volume of the doughnut." My non-rigorous reasoning was that the outermost "dx" would stretch by the same amount that the innermost "dx" would compress. I dunno if it works exactly that way, but the cylinder idea at the end of the video felt similar to my thoughts.
That's exactly what the theorem ends up doing.
For the confused ones, Pappus' theorem is also known as Guldin's theorem.
I get a giant coffee cup (although I'd have to settle for a glass coffee pot since giant coffee cups are hard to find), I fill it half way with coffee that I've measured. I then take the donut and put it into the coffee and push it down and I measure how much the coffee has risen in the glass coffee pot. I then compute the volume by: measured height difference times PI times square of the Giant Coffee Cup Radius. That gives me the volume. Another way to do it is to measure the thickness of the donut ring, and the inner and outer radius and use a little calculus. But I like the Giant Coffee Cup method better, because that way I can drink the coffee and eat the soggy donut. So why this method better; besides having breakfast? Because from a Topological point of view: there isn't any difference between a coffee cup and a donut. Works better on an old dried out Donut. It tastes better too and having food and drink and not an empty stomach helps when doing math.
What about a value of a random rock using calculus?
What about the volume of a donut using the water method (first discovered by Archimedes)?
Definitely not water. Maybe coffee is okay. 😆
water? custard!
won't work, the donut will soak up the liquid ruin the result 😜
it will be more accurate because a donut is not a perfect Taurus and there is air in the donut that you might not want to include.
It's winter here in Brazil
National donut day is a Friday in June so it's appropriate.
can u explain how 3.25 came in circle
Measuring in centimeters? Thought you were in the US where they will use anything to avoid metric!
Hahaha
he seems to be an abnormal
The last method was definitely the best, since you didn't have to use integration!
Well, since a donut is a circle, you could calculate the volume of the circle and then subtract it from the hollow circle in the middle of the donut
"Volume of a circle" is meaningless, since a circle is a 2D shape, and volume is a 3D concept.
I will just say.....maths is smthg which is endless🥰
W professor
Washer: π*(r)^2
Shell: 2π*r*h
Just some notes for me
Can't we do it like taking a circular part (DX) from the ring and then then find its area and integrate the whole thing according to the circumference of a circular so that it adds up as the volume .
I tried it and gues what ... It worked as was much easier then your method (yours is a bit lengthy )
But you was more algebric .
Integrals: Am I a joke to you?