Find the Volume of Any Shape Using Calculus

Calculus isn't just abstract mathematics, it is an incredibly useful tool. Here I show you how to use it to derive the volumes of 3D shapes. Check out my posters here store.dftba.com/collections/d...
This was the first example that really opened my eyes to the real value of calculus. I learned it in university in my physics department mathematics class, and it was just the beginning of a long journey of me wielding calculus at all sorts of practical problems in physics, and it gave me the power to see where the equations of physics came from, rather than just having to learn an memorize them. Awesome!
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[@mr.b1130]

"This is Math; we can do whatever we like." -somewhere around 5:00

[@adibmukhlis5256]

4:55 *

[@cory99998]

This actually helped me wrap my head around calculus quite a bit. I understood the general concept but seeing it actually worked out this way makes way more intuitive sense than someone throwing a bunch of equations I've never heard of at me.

[@BassDropper991]

Nnb bn nni. Bonnbb inb bnbnin😊bi. N binnnbnnn ni nnbin innnnninnnbnnnb. Bbbnnbii😊nn nn nn

[@BassDropper991]

B uuibbbbojubb n bbbbbb. Bbbbb. Bbbbbb bj

[@joeymorangarza]

"When you hit those infinities, that's when your approximation becomes exact." 1:36 The only reason to learn calculus is so you can say things like this.

[@metakatana]

It comes from Real Analysis.

[@silverspringer3577]

@@metakatana What do you mean

[@somebodyiusedtoknow2012]

DynastyGuy big brain math

[@silverspringer3577]

@@somebodyiusedtoknow2012 No I mean why does it come from real analysis more so than calculus, limits are literally one of the cornerstones of calculus

[@metakatana]

@@silverspringer3577 The limits used in calculus are from Weierstrass's rigorous definition of a limit. Weierstrass is considered to be the father of modern analysis.

[@yutharsansivabalan8285]

Learning calculus during lockdown, Great timing

[@MG-hi9sh]

Nice one, great to see people making good use of lockdown.

[@lautarorivero7799]

this is what i did during last three months hahaha

[@mark0032]

Its how Newton invented it.

[@sakibhasan7857]

U remind me 3blue1brown

[@SYFTV1]

@Gente que le encanta estar mamando

[@HugeRademaker]

I so love your videos! Thank you. Geometric speaking: a pyramid always has a squared base. The shape with the triangular base is a tetrahedron.

[@UltimateSuperSaiyan]

you can also take a triangle through a y=mx+c function and then a volume of revolution of 360° about the x axis.

[@fkncompton7124]

Wouldn't it have to rotate about the y axis

[@9308323]

​@@fkncompton7124 Since the volume of solid shapes don't care about orientation, so is using either method. Of course, you'll need to get your values and equations right in the first place but that's a given.

[@abusoumaya8469]

i 'm doing it like that. more simple

[@MathCuriousity]

My trig precalc knowledge is weal but can you tell me what the formula would look like? Do we need to use unit circle?

[@MathCuriousity]

Can you explain what you mean by taking a triangle “through” y=mx + b?

[@nac9880]

Find the value of any shape with a rectangle: 1. Make an exact replica of the figure 2. Fulfill it with water 3. Make a rectangle replica that fits exactly all the water 4. Calculate the area of the rectangle

[@stargazer4574]

Well that's easy.

[@patrykkwoczak2114]

StarGazer45 Great idea! Even easier: just get any old rectangular box that can hold the amount of water, then just calculate the area of that base * height that the water fills!

[@Brunoenribeiro]

It makes sense, but I think the beauty of math is that you can find out the volume of ANY shape. What if you want to calculate the volume of a swimming pool with a weird shape? The volume of a cloud? Or of an entire planet? All you need is math and imagination :) I'm not underestimating your comment, of course. Just sharing other way to see it

[@kseriousr]

Erm, I think you meant 'volume'.

[@9308323]

If you can't fulfill the water, you can also satisfy it.

[@ranadeep7462]

IMO the expression for non inverted cone should be r/R = 1 - (h/H)

[@legiavu6398]

Ranadeep i thought the same

[@brogant6793]

Ranadeep his formula doesn’t give r=0 for h=H so it isn’t correct it gives r=HR-R which is wrong (unless I’m missing something here)

[@gunthermathiasbausellapiz5211]

@@brogant6793 Exactly, must use the one from Ranadeep

[@ranadeep7462]

@@brogant6793 IMO u r not missing anything. I think it might be innocent mistake on his part while he was explaining the relationship between r and h. This thing can happen to anyone.

[@Brunoenribeiro]

true!

[@mendbayasgalanlkhagvadorj3097]

Man keep this up, I’ve been learning a lot from this channel 👍

[@galzajc1257]

Great video and one of my favourite pieces of math. It's also very related to my last years high school research project, where i found an elementary(using just high school math) way to derive formulas for volunes of all regular polytopes and 1 formula, that directly gives volumes of all platonic and archimedean solids except the snub cube and snub dodecahedron, that were a little trickier. Moments of inertia are even more fun.

[@danielmahmoudi6731]

A student asked me a question about a tetrahedron and figuring out a proof for the varying height is exactly what I needed. Thank you! I've been working on this for a few days on and off.

[@planetsharks4746]

I've been watching your videos since I was 10. I've now revisited your channel again and it makes me so happy you're still making great content. You've made an impact in my life and I want to say thank you.

[@domainofscience]

Ah that's so nice of you to say. Thank you!

[@samuraijosh1595]

@@domainofscience At 6:51 shouldn't it be r/R = (1-h/H) I'm so confused...😭😭

[@Jonas-gm4my]

@@samuraijosh1595 if R =1 and H=1 then you would be right. But he just used some value to better picture what he was doing.

[@pranavpolakam5371]

@@samuraijosh1595 I think so too; I’m not sure if we understand the statement though. The reason I think it’s supposed to be what you said is because r/R is a ratio. So if we are working with the 0.25 and 0.75 example again, you would need to do smth like 0.25 = 1 - 0.75. Since r/R and h/H can both theoretically never exceed 1, that works. Or at least that’s what I thought. The only way to rationalize this as an incorrect solution from what I can tell is that this isn’t what he’s solving for. There’s a chance we misunderstood what he’s doing on a fundamental level.

[@ThePersonWhoKnocks]

I really appreciate this video! As someone who struggled with calculus in school, this breakdown of how to find the volume of any shape using calculus is incredibly helpful. The way the author explains the concepts is so clear and easy to follow. I particularly enjoyed the explanation about how hitting infinities is when the approximation becomes exact. It's amazing how math works! I also love the comment about how the only reason to learn calculus is so you can say things like that. Overall, this video has helped me understand calculus a lot better, and I can't wait to apply these concepts in real-life situations. Thank you, Domain of Science, for another excellent video!

[@jbw6823]

When I was in high school, this was one of my fav things about calculus. I was like " why didnt you teach us this earlier!".

[@musclechicken9036]

you can also use H and R to form a linear function which have the x and y intercepts of the values R and H, then take the volume of revolution by integrating the linear function squared then multiplying it by PI for the radius. Though, the way you taught it is way more intuitive and beginner friendly. It helped me learn more of what an integral really is, thank you!

[@ramonasosna]

I only recently discovered your account on KZbin and it has really helped me re open my mind to the world of maths and physics as this is something I want to study, you explain things very well and I thank you for doing what you do 🎉😊

[@sessybish1001]

Great video sir! Its rare that by the end of a math video I say, "that is cool." Inspiration for me, thank you.

[@guillermocasanovaaguilar8180]

The exercise is very well explained. I want to make a contribution: The expression of r as a function of h comes from the fact that the triangle formed by the segments h and r is similar to the triangle formed by the segments H and R. The triangles are in the position of Thales's first theorem, so the expression comes from the similarity of triangles, and then it is automatically true that h/H = r/R. I explain this because it may be a bit difficult for someone to understand the proportional relationship between the triangles. Greetings to all.

[@domb02omkarp32]

Thanks a lot sir

[@Stephendsilva1]

I dont understand - if R and H are 4, then r is 1 then shouldnt h be 3? How come they are equal?

[@smiley235]

This was great, actual practical application to really drill a few things home.

[@snyperbro9502]

At 6:38: shouldnt the formula be r/R = 1 - h/H?

[@nanak3363]

H/H-h/H = 1-h/H . @snyper BRO

[@haileywarner5109]

Nanak its not H/H - h/H, just H - h/H

[@voom6996]

@Nova Flares yeah but if the height is zero, the answer should be R and not RH (which isn't even a length anymore), it has to be r(h)=R*(1-h/H) which fits more with his explanation. When h is 25% of H, then r is going to be 1-25%=75% of R and vice versa.

[@vitaminaccc]

@Nova Flares He said at 5:50, thanks, I was having the same question. Time-saving.

[@fiNitEarth]

Had the same thought. Yes I guess so, since when h=H, r/R should equal 0, and that wouldn't be the case when H isn't 0 if you're using the formula from the video

[@MindMathMoney]

How could I not have heard of this channel?! Im in *LOVE* ! 🥰🥰

[@danielrybuk1905]

Instead of writing r=Rh/H i think it is simpler to write r=h×C, where C is just a constant. Integration looks the same because C is a constant, and u just plug in C=r/h in the end. I also rotated the "triangle" (the side view) 90° so it actually looks like inspecting a function over h, but that probably just personal choice. Great video, i liked it a lot, and u are right, this sort of question is very good to inspire thought and understand calc.

[@5_Sigma]

My approach for the sphere was the formula for a circle in two dimensions, so r^2 + h^2 = H^2. The rest was just like the pyramide, but times two, because there are two half spheres in a sphere.

[@Brunoenribeiro]

Thanks for helping more people get to appreciate how calculus is smart and beautiful

[@theviswaskoundinyan536]

one of the best video i ever seen! do more videos like this!

[@herp_derpingson]

I remember doing this back in high school. It was magical when I did it the first time. I did it for torus too then verified the answer with Wikipedia.

[@jayantmandal712]

Can u derive the surface area of torus?

[@FAnizah_edits29]

wow this helped a lot, I have already bought some of ur posters ( the physics, chemistry, math, donut of knowledge, chemistry, and math notation, lol I love ur vids )

[@SonuKumar-fm3jn]

Glad KZbin recommended me this video ❤

[@tectzas]

So clear and awesome!

[@richardslater677]

I’ve tried to understand calculus using KZbin. Honestly, this is the first video I’ve seen that tells us what calculus actually does.

[@marco.nascimento]

This is awesome, great work

[@user-ii3nh7wi2w]

Nice explanation, Thanks!

[@longstoryshort-biochemistr1172]

Great Job!

[@TheScienceBiome]

Wow, this is pretty beautiful!!

[@bradleybeauclair8282]

Same thing except check this out: For any related rates problem go to the top right corner of your notebook paper and write this down: Height 1, Height 2, Height 3, Change in Height 1, Change in Height 2, Change in Height 3. Or even better: H1, H2, H3, dH1, dH2, dH3. You are given a height. You are asked to find a change in height. You usually need to solve for a height using pythagorean theorm if for example, it's the ladder problem (2D object means no H3 and so no dH3). I started doing my calc 1 that way over a decade ago and I received a complement from my professor. She had never seen that before. The faculty went around to try and find who taught me that and all they could figure out is one of the professors had seen it once before, when she was in college in the late 1970s, One of the grad student teaching assistants did that. It's the superior way to do Calc 1 and I'm the only person who I know personally who does it that way. Go write down what I just told you to write down and you'll see why immeditately. It should look like how you would write sin,cos,tan, and then to the right of those three; csc,sec,cot but H1,H2,H3, dH1,dH2,dH3

[@jalipathak3156]

" Beautiful "

[@freemusicscore7281]

Thanks very ensightful! BTW 08:19 has to do with Triangle Proportionality Theorem right?

[@MichaelHazell]

Nice! At first I was thinking that you were going to take the solid of revolution route but this works as well :)

[@MathCuriousity]

My trig precalc knowledge is weal but can you tell me what the formula would look like? Do we need to use unit circle?

[@fred5663]

Hey man nice video. How is the name of the operation you've done with the r/R = h/H ? What's the logic behind that?

[@kingsleychen5222]

He has made a simple idea into a complicated one

[@davidaugustofc2574]

An inconvenient into a possible one

[@illumexhisoka6181]

That's math

[@nathaenwanta6452]

It’s a basic calculus 1 problem

[@khiemgom]

Well u can actually derive it non-calculusly but that is shitty as hell and require like 40 min proof so not anywhere simpler

[@multimate_]

@@davidaugustofc2574 inconvenient isn’t impossible in the first place

[@OWASIA]

Sir,your video impressive. Can you guide. How can I use in finding volumes and areas of my daily life problems. Or some other work using this calculus

[@ahmedbyahmed4405]

Nice topic I like your videos continue 👏🔥

[@j.pesquera]

This was incredibly thorough and insightful, this is basically a representation of how to actually *DO* calculus. Not just read a textbook, memorize formulas, and regurgitate them on an exam.

[@sairoof4704]

Cool And the less weird shaped 3 makes it 10 times better

[@ramkishansharma8207]

Which software are using for video and animation for figure

[@mynameisawesomeman]

to be 100% clear to viewers, you should include the missing index in your finite sum, as well as a definition of the sequence {h_i} for i = 1 to N, since until you get to the definition of the integral, it's not obvious that delta(h) stays constant (for any given sum) while h itself is changing (getting smaller as r gets smaller, as we move up the to the peak of the shape). In other words, work out the finite sum solution using a numerical example for some chosen delta(h) and choice of N.

[@MathCuriousity]

Can you explain it like im five? Having trouble Following what your caveat is.

[@ericgomes6290]

thank you so much for the video....

[@kaitlyn__L]

The Egyptians used the first couple of steps of this method to work out the internal volume of their pyramids.. I think they turned it into a right angle triangle by moving the slices to left-align.

[@themeralderp9615]

I tried to get a solution using the first equation, but I didn’t get the right results. Here is what I did: First you would divide both terms by R to make r = R(H-(h/H)), then you would multiply by H to get (R/H)((H^2)-h)). Now you would create constants to make it more manageable: a = R/H and b = H^2 results in the final equation, r = a(b-h). Since we are going to square this, we will instead have a^2 = (R^2)/(H^2). For the second term, squaring gives (b^2)-2bh+(h^2). The constant (a^2) would be moved to the outside in integration, whereas we will have to integrate the other term. Using the power rule, we get (b^2)H - b(H^2) + ((H^3)/3). Expanding and simplifying constant a and b results in pi*(R^2)*((H^4/H) - (H^2) + (H/3)). This results in pi*(R^2)H*(1/3) * (3(H^2) - 3H + 1), which is not the intended result. WolframAlpha gets the same result, so I don’t think I integrated incorrectly. Any ideas?

[@netad7771]

Dead useful vid, thanks mister. keep gooing 🎉🎉🎉

[@sayantan2201]

Of the top of my head, cylinder is base x height, cone and pyramids are 1/3(base x height) , circle is 4/3*pi*r^3

[@jakobion9216]

I really like a soft music in the back, helps me focus

[@samuela883]

Superb! 👍

[@straaths]

Thank you

[@bene5776]

For a sphere, what is the relationship between r and h? Is it r = sqrt(R^2 - h^2)

[@sagnarte6074]

So cool!!!

[@hosseinfaridnasr2778]

Great vid! You could also use the 30, 60, 90 triangle to find the relationship between r and h.

[@samuraijosh1595]

Lol, no. Not all of this are 30 60 90 ones....

[@MathCuriousity]

Can you explain why not? If he didnt use 30 60 90, how did he even know the relationship between r and h right from beginning when discussing “ half way up h is half way r” im paraphrasing.

[@eddyboyblues]

Awesome thank you.

[@divij1937]

Hey, could you please post/do a video on map of scientific research? Thanks for the other map videos you guys have posted!

[@billscoaching9711]

Great video 🙂👍

[@g_rr_tt]

What is the general procedure to find the volume of a general 3D solid shape that isn't symmetrical about any axis? (i.e. just a blob in space)

[@Juliandav0908]

Amazing stuff

[@TOMTOM-nh3nl]

Thank You

[@lastchance8142]

The real miracle of Calculus is how Liebniz and Newton developed it at the same time independently!

[@amorrisphoto]

Love how you explain things. What kind of pencil is that? It's gorgeous.

[@LusianRoth]

I think it's MUJI Hexagonal Wooden Pencil

[@tonymoh6918]

What separates r from dr and h from dh? Aren't they both smaller pieces of the larger constant: R and H? Why not do the sum of pi(dr)^2dh?

[@reagandoyon]

just a little confused,,, how do you know that at half the height of the cone it will be half the radius? doesn’t this idea only apply to this specific cone with a gradient of 1? what if a cone has a different steepness with a gradient of 2 or 1/2? how would you go about adjusting the width of each disk as the cone thins out? or is a gradient of 1/1 part of the definition of a cone? thanks!

[@persassy7076]

The easiest way to imagine this (imo) is with similar triangles. We have a similar right triangle for every triangle with r smaller than R. Each of these triangles is within the initial triangle sharing 2 sides (the definition of a similar triangle). So it follows that r/R = h/H, as it does in any similar triangle where the proportion of height to the base is preserved.

[@midnightstorm4290]

Is there any point in doing this in a GCSE exam? Or should I remember the formulas for method marks .3.

[@Quotenbrtchen]

Spherical coordinates for the win! (when calculating the volume of a sphere)

[@mirijason]

no need, the volume of the sphere is 0

[@Quotenbrtchen]

@@mirijason Sphere, ball, to-mah-to, tomato

[@myshowsanchez]

Seeing the illustrations of shapes is so cool. I wish i understood a quarter of what he’s explaining. I extol mathematicians.

[@Jaysin999]

Opened my eyes to help with optimization problems, but my prof said my algebra is lacking and😢 rip lol, ill have to retake calc again. I can say i felt a bit motivated to do calculus after watching this, but its the weekend rn

[@romailto9299]

Sorry for a belated question, but when you set up the equation r/R =... should it not have been r/R = (H-h)/H = 1 - h/H ?

[@cmmndrblu]

I love all of these videos. My problem has always been the cognitive load of language. Listening to "as delta h gets close to zero" I realise that the image behind this language is math focussed. These terms have low cognitive load for the speaker. I just think of printing a shape in slices over time. When there is no change in the slices you get a cylinder. When there are changes in the slices you get sth else. But the changes can be categorised. If you print a shape from 0 to completion and you alter the area of the slices at a consistent positive rate, you get a disappearing cylinder like a pyramid. But you could do other things like consistently twisting about a centre. When I see it as a series of frames it's far easier to understand, and once I understand the concept the label for it can be arbitrary but we use agreed labels so we can speak and understand each other. I have always hated the tendency towards Latin and Greek words in our naming of things because they sound innately complex in English (but not so for example in Spanish). The word "integral" has one meaning as a noun and another different one as an adjective (crucial), but the word "whole" does not.

[@turdyhole1233]

How do you write your symbols so nicely??

[@muhammadyameen4998]

hhhhhhhhh the less weird 3, I was looking your way of teaching. thanks alot.

[@shermanthompson871]

What about a solid formed by the Dirichlet function +1 on top from 0 to 1 and the x axis on the bottom? No measure theory (I don’t count that as calculus hehehe.)

[@nofacee94]

Your link to the store is broken...

[@ItzSoulGD]

It looks cool

[@ryanperry7673]

You can also do this with double integrals

[@siddharthsambamoorthy4479]

What abt more complex and irregular shapes?

[@shadrachhamner1439]

Very tricky setting up a ration to represent the relationship between radius and height

[@MathCuriousity]

Why is this tricky?

[@HoSza1]

This idea works only for a smaller class of all possible shapes (except when one defines shapes using boring functions only, i.e. smooth, integrable).

[@Tomohiko_JPN_1868]

My idea for SurfaceArea (Sphere). (i know , it is illogical, but works on somewhere). What is the simplest object in 3 Dimension world ? it is a Triangular, has only 4 vertexes and 4 sides. ..... SurfaceArea(Triangular) = Area(Polygon of tri) x4 sides. Now, we think, this most simple object as a kind of "neutral element" in 3D. Next, Sphere is also a simplest object in 3D. So, we can think it is a variant of "neutral element", = a variant of Triangular ( or Philosophical Triangular). So, it must have 4 sides. ..... SurfaceArea(Sphere) = Area(Polygon of circle) x4 sides. --> (pi x r^2) x 4

[@h1a8]

It's funny how using cylindrical cross sections approximate (converge to) the volume of a cone (and other objects) but do not approximate surface area. You need frustums to do that.

[@leontommy9511]

what mechanical pencil youre using?

[@bajonk9024]

My man just explain high school student calculus better than their teacher

[@ScourgeofBabylon]

I don't understand how you got the relation between r and h to be the 1:1 ratio you have here. What is that based on?

[@markmpm]

Please, always list the music played in the video! (With links if possible.)

[@user-py3vs6ox4y]

شكرا جزيلا

[@damodaraomalley3974]

BRILLIANT... so the answer still looks like hieroglyphs... what is the volume in cm2?

[@stargazer4574]

You know you're early when there are no views and no likes.

[@appleslover]

No shit, Sherlock

[@chrissmith1152]

wow I never knew that, thanks

[@Shubham_pandey-nk1un]

@@chrissmith1152 🤣🤣

[@alvinkwok587]

then what about objects with more complex shapes like tetrahedral, trigonal biprymidal

[@boramenderes3272]

Guys is there a way of integration to calculate volume of triangular based pyramid? I am tring to integrate for 4-5 days, I will be crazy. Just for asking btw

[@Neshx1]

love your pencil

[@darwinvironomy3538]

i can understand it, but i don't know how we can find the relationship of two variables using MATH, I Thought hypotenuse line is just describe as y=x why Why a ratio?

[@Synthenist]

Wow beautiful

[@JavierMartinez-ol9uu]

Please make the Map of Philosophy

[@Reseng0411]

Cool man

[@Stephendsilva1]

Doubt- if R and H are 4, and r is 1 then h should be 3. For eg. How come r/R = h/H? Please help

[@andunyaa]

Nice