Area of Surfaces of Revolution | Derivation & Example

If we revolve a curve around an axis it forms a surface. We can use Calculus to compute the area of this surface, much as in Calculus I we computed the volume contained in this region. The derivation of this formula works in much the same way.
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[@funwithwaniarumaisaareeba7586]

No one in the whole KZbin has explained this concept as much clear as you did. Hats off to you. Really deserve much more views.

[@firsttnamee3883]

yes

[@kimberlyann1022]

I cannot agree more with this. I watched all of his ads as my thank you

[@arnavbhavsar8567]

This guy is definitely the best teacher on youtube.

[@deyomash]

So, funny, but. I'm a graduate theoretical physicist but until now i have not understoof why in the AREA you need to take the arclength but in the volume you're okay using just "dx" (just pi * f(x) ^2 dx) even though the little cylinders are still "curved"..

[@DrTrefor]

this happens so often that a little thing from calculus or whatever is missed, happened to me all the time in grad school, like how did I not know this?

[@deyomash]

@@DrTrefor no you dont understand, i literally dont know still. Pls tell me hahaha

[@moazamkhan]

Yes WTF

[@Areyouserious1]

my doubt is why dont we use the arc length formula when finding the volume generated by the revolving function....There we only take the width as dx

[@firsttnamee3883]

the most underrated math channel in youtube

[@stephaniegarcia8557]

Just found your channel and all I can say is WOW. I love the way you teach! It's very clear and easy to follow along. Definitely subscribing!

[@js7244]

Thanks Trefor!! once again you are saving me so much time and giving me real insight in to the subject

[@angelfernandez4235]

Really great explanation & illustration! Had so much problem imagining it but the way u visualised it rly help,thanks a lot!

[@madhavanand756]

It's beautiful the way you have explained, directly goes in my head. And it is totally unique the way you explained

[@zihuiwang8649]

Thank you! Really helped my maths project at school!

[@SpazzHobo]

The visual made it very helpful for me. The trick is to relate the surface area here to the idea of how the surface area of a cylinder is derived from a rectangle. That idea combined with your visual made it clear for me. Thanks!

[@eyadnasir6802]

Amazing! Thank you for a unique way of explaining this.

[@felipeseixas7028]

We are watching a master at work 😎

[@aslpelinsuozturk3106]

You deserve a lot more views. This is an amazing video.

[@subhradeepghosh2523]

I was struggling with the derivation. This video helped me a lot. Thank you.

[@nazihnassar2101]

the best video on the internet explaining this topic p.s you saved my life

[@alisapuskala1437]

Super clear explanation, and an intuitive explanation. Good material for an ia

[@royalzak4019]

Your teaching style is awesome Love from India.

[@leadersheir9377]

"So in place of the width, I'm gonna come along here and put in that square root of *all that nonsense*" 🤣🤣🤣 Absolutely LOVE your content, professor!

[@IlIlllIlll]

That really helped, thx🙏🙏

[@adnanmohamed6517]

8:28 minutes is way more better than 1 h lecture. ( I don't know who said to these professors that math is just solving examples using formulas that you don't know from which hell they came from) Thanks alot

[@renefsierra]

Thanks, great explanation

[@cesarreategui2693]

THIS GUY IS THE BEST!!!!!!

[@lukaskamin755]

Cool explanation. Thumbs up 😀

[@ananyaprakash4262]

Your videos are so well done🤗

[@hmudisalam]

excellent !!

[@iknowimreal]

Thank you

[@DrTrefor]

Glad it helped!

[@albertmendoza1468]

Understandable. Have a nice day!

[@treasuresoffaith7027]

I really like your videos. 👍

[@peon3715]

Just amazing. I subscribed inmediatly, I don´t know why he doesn´t have more subscribers.

[@DrTrefor]

Haha right!?

[@kin_1997]

thank you for this :)

[@nicholasbolte2248]

enjoyed the video, was just wondering what software you use to show the 3D model (including the z axis)

[@tiagoantonio2343]

fantastic

[@davidhofmann4857]

You are an awesome teacher, thanks!

[@DrTrefor]

Glad you think so!

[@fernandotanase114]

Could you also link the mentioned videos in the description box, please. It'll make it easier for students that have multiple browser tabs open and trying to understand the concepts while watching the videos.

[@Pranav_prince]

Superb explanation and understood well in single watch

[@DrTrefor]

Glad it was helpful!

[@anaidamadhavan1271]

You are the best man. I understood this so much clearly. Thanks alot🤗🤗

[@DrTrefor]

Happy to help!

[@anaidamadhavan1271]

@@DrTrefor subscribed 😊

[@rfc7643]

why we can't find area of surface only by circumferences without width?

[@prateeksingh974]

Too good

[@siddharthkhandelwal933]

Thanks!

[@DrTrefor]

Thank you so much, you're amazing!

[@fan-gq8lj]

life saver

[@ashishKjr]

Initially you say x_i * is some point in the partition (and I assume by some, you mean arbitrary) but later it seems like x_i * depends upon ∆y_i and ∆x_i by mean value theorem. So, doesn't x_i* depends upon the partition and is not arbitrary?

[@shreeramshreyas9761]

U d best man !!! I have a small doubt Tho. Why can't we consider the ribbon as an cylinder instead of a circle ?

[@toomanycharacter]

2:51 We're not going to worry about the difference between the bigger and smaller radii, because in the limit it's going to be smaller and smaller 3:43 But we ARE going to worry about the difference between the "Width?" and dx, despite the fact that in the limit it's going to be smaller and smaller? Why the distinction? I mean, if we would decide not to worry about both of those small differences, then we would get 2pi * f(x) * dx as the surface area. If we decide to worry about both of those small differences, then I don't even know how we would get any surface area at all. Clearly, the distinction is very important, but why did we decide to make it?

[@toomanycharacter]

I think I kinda get it now. The absolute difference between the radii is going to be smaller and smaller, and the relative difference between them is going to be closer and closer to 1. So in the limit, the two radii will be equal in both ways. The absolute difference between the "Width?" and dx is going to be smaller and smaller, but the relative difference between them is not going to be closer and closer to 1 - it's going to be closer and closer to sqrt(1 + (f'(x))^2). So in the limit, as dx and "Width?" approach zero, "Width?" is going to be a constant multiple of dx, so the widths will not be equal in both ways. Because the radii are equal in both ways, we can think of the surface area as the surface area of a cylinder, with the "Width?" being the cylinders width. In a strange way, in the limit, "Width?" becomes parallel to the x-axis while remaining a constant multiple of dx. Is my interpretation alright?

[@dudeiknowman]

​@@toomanycharacter i had the exact same question as the original comment, and your interpretation feels insightful. In that strange way, in the limit, the width is parallel to the x axis, but remains some constant multiple of dx. Wish we could get professor Bazett's thoughts on it.

[@mnada72]

0:50 Do you have a video on this topic in the Calculus I playlist?

[@DrTrefor]

This is part of the Calculus II playlist as I have it, but each college or university does it a tad differently.

[@mnada72]

@@DrTrefor Thank you, what I mean is the calculation of a volume enclosed by a curve I couldn't find a video on either calculus I or II in the playlist .

[@mellifluousb.4549]

Doesn't the formula need an absolute value for f(x)? I mean if the curve we examine is f(x) = -x^3 with 0

[@gaureshnaik7248]

Sir u teach so well 😊😊

[@DrTrefor]

Thank you!

[@willthecat3861]

It was all good... particularly the invocation of the 'mean value theorem ' to justify using the derivative... it was all good until the end when he didn't point out that... except for some trivial families of 2-D curves... the integrals can be very difficult to evaluate non-numerically. Profs always choose an example your cat can calculate, and then leave you to burn your brain trying to do ones that require a bag of obscure integral tricks.

[@dvsalim09]

Hello Dr. Trefor, I really find your videos helpful. Just need to know, how can I cite your videos?

[@DrTrefor]

A link is fine!

[@dvsalim09]

@@DrTrefor Thank you

[@thezoz9476]

Why isnt the width dx

[@user-uh8wk5uy9n]

Find the surface area due to the rotation of the area between f(x)=x^3 and g(x)=x

[@MrMarkgyuro]

dear Trefor why we divide 2phi by 36 in the last part?

[@MrMarkgyuro]

@@DrTrefor so we solve for 1/36. Like 1/36 * du/dx = x^3? im lost :)

[@MrMarkgyuro]

@@DrTrefor no i see sorry :D du/36 = x^3 dx..

[@ezequielsm]

thankyouthankyouthankyouthankyou

[@BilalAhmed-on4kd]

why don't we think of it as finding the sum of the curved surface areas of infinitely many cylinders, since the width is infinitesimally small, it's curvature should not matter, just like in volume

[@zechang573]

😊

[@zechang573]

thanks for teaching

[@amaljeevk3950]

🥰

[@jansmith7387]

Why is my answer negative?

[@suleymanemirakin]

still dont understand why we dont take delta x

[@luisramos3748]

circunference....

[@user-fn7vs6pb6n]

For those looking for Arclength Formula video, here you go kzbin.info/www/bejne/pnmQgKCgYsiApc0.

[@logical29]

You talk like Krieger from ArcherFX

[@marklourencelucas9495]

Goodbye organic chemistry

[@natty1]

no idea what just happened im just dumb

[@maxjang2446]

hey man, thank you for saving my ass for a math investigation paper🫶