Surface Area of Revolution By Integration Explained, Calculus Problems, Integral Formula, Examples

This calculus video tutorial explains how to find the surface area of revolution by integration. It provides plenty of examples and practice problems finding the surface area generated by a region or curve when rotating about the x axis or y terms. This video provides the two integral formulas that you need and how describes how to find the radius in terms of x or y variables. Integration techniques needed to solve problems include u-substitution , simplifying radicals, and factoring perfect square trinomials.
Antiderivatives:
• Antiderivatives
U-Substitution - Indefinite Integrals:
• How To Integrate Using...
1st Order Differential Equations:
• Separable First Order ...
Initial Value Problem:
• Initial Value Problem
Area Between Two Curves:
• Area Between Two Curves
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Disk and Washer Method:
• Disk & Washer Method -...
Volume By The Shell Method:
• Shell Method - Volume ...
Volume By Cross Sections:
• Volumes Using Cross Se...
Arc Length Calculus Problems:
• Arc Length Calculus Pr...
Surface Area of Revolution:
• Surface Area of Revolu...
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Work Problems - Calculus:
• Work Problems - Calculus
Integration By Parts:
• Integration By Parts
Trigonometric Integrals:
• Trigonometric Integrals
Trigonometric Substitution:
• Trigonometric Substitu...
Integration By Partial Fractions:
• Integration By Partial...
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[@TheOrganicChemistryTutor]

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Thanks Jessica

[@georgesadler7830]

Professor Organic Chemistry Tutor, thank you for another incredible video/lecture on Finding the Surface Area of Revolution in Calculus. I had problems with this topic in years past, however by doing a ton of problems and watching videos, my understanding of this material has drastically improved. This is an error free video/lecture on KZbin TV with the Organic Chemistry Tutor.

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[@lilyfranks9839]

for the time 16:23 and on why is R(y) = y and not the whole x= equation that is given?

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i was thinking the same thing.

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mystery unsolved after years

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[@ah.a2466]

I'm sure he made a mistake because f(y) is not equal to fx and he wants to flip around the x axis so he needs fx.

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[@francovarela4062]

I cannot find videos about definite integrals involving the Surface Area of Revolution. What if you have 0

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the semicircle example is just genius

[@joshuajackson1998]

Is R(x) or R(y) always parallel to the curve (or we want it to be /manipulate function)? I’m having difficulty understanding R(x) and R(y) so when it is equal to the given function or just equal to “x” or “y”

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[@christopherdeckinga2421]

At minute 4:17, you change the limits to [1, 145]. Correct me if I am wrong, that is done to allow us to work with smaller numbers... and it is not absolutely necessary to find the correct answer? Do you have any videos that focus on that technique? Ive seen it done in my calculus book, and Ive seen you do it a number of times, but Ive never been able to master it. Lastly, if you are looking for more topics to cover in your videos, there is a book used for "Applied Calculus II" called Basic Technical Mathematics with Calculus (by Washington). There are a few concepts in that book that I have only seen covered by a few videos; namely Fourier Transforms, if I remember correctly. Thanks so much for all the videos you've created. I know you are moving towards paywalls, etc. So be it, these videos are gold... absolute gold.

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[@eddied4969]

Question starting at 12:38. My calc book says when you revolve about the x axis. The formula is S = | a to b (2pi (y) sqrt (1+(dy/dx)^2 ) dx = | a to b 2pi f(x) sqrt (1 + [f'(x)]^2 dx which is different than the way you figured it out. The answer looks correct just wish i understood why..

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[@_Parachini_]

Why did you change the interval of integration around 4:50? If this is a common practice could someone tell me which video to watch. Thank you btw your videos are god sent.

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Need questions with integrals involving trig and/or logarithmic functions.

[@Chrisymcmb]

I don't quite understand why we do not need to somehow substitute back in the value of the function for the radius when the solid is rotating on the x-axis but we are solving for y. If the function is x =*****y, would not the value of r(y) be *****y, instead of just r=y ?

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[@ThOtherzz_]

Why you always change the limits (e.g. a=1 and b=2) if you're going to use u-sub?

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Thank you Sir

[@abiryaakovalmoznino3395]

Hey is it possible to upload a calc 3 video about surface integrals?

[@aryamandarda7654]

Hey I was wondering if you could tell me how to derive the general surface area formula. (S = integral from a to b * 2pi * f(x) * Squareroot(1 + (dy/dx)^2) dx)

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3 years later.. did u ever come to understand it? I am there now lol

[@michaljhon6375]

Sir After substitution what u did ?? and how limits are changed to 145 and 10

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[@cuthbertlwinga4300]

Thank you.

[@ambercao6637]

I have a question, Why do u change the intervals in some of the examples but don't change it in other ones?

[@Bella-ul1fq]

I know this is 2 years late, but you only change the limits when you use u-substitution to integrate :)

[@alisherzinullayev5026]

Does anyone know why he used dy in the third problem if the task requires rotation around the x axis?

[@beetogarcia9983]

At 6:50 Wouldn’t it also be possible to square the number then cube it?

[@jamesfredrick612]

Legend ❤

[@brandonhung9990]

Mastering volumes of revolution really helps before watching this video

[@Chrisymcmb]

I'm curious as if to anyone tried solving the problem at 7:05 using U-Substitution. I tried and what is weird is that when I get to the steps of changing my bounds since I performed U-Substitution, my new bounds end up being the same [Lower Bounds: 3, Upper Bound: 3] I'm wondering if this happened to anyone else, and if there was a workaround about this.

[@raze337]

when rotated at x axis, is the strip dy? When rotated at y axis the strip is dx? please enlighten me, thank you

[@triparadox.c]

13:00 If the video had stopped there, I knew that you are my h.s. teacher

[@bentsporks4339]

slight anxiety from that

[@erikmachado4489]

I believe there was a mistake at 11:18. You forgot to distribute the negative the -x^2 which would make the answer sqroot 4-2x^2

[@nightlymerecat6519]

I know I am no expert but you could have made DU/ 36= X^3 dx canceling out the x^3 as you replace DX. Just what I do to make things simpler and clean up my work

[@anoopgill7245]

At 18:00 why didn't we change our bounds like in the previous problem?

[@bman9469]

Does the radius and f prime both have to be in the same terms? (X's or y's)

[@Mark_Hoag]

Pretty much, because you have to integrate. Not saying there's not a case that wouldn't work

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this is just how to solve these problems not to teach or explain the concepts, that’s for your professor/ if you buy his course

[@adamruggieri3447]

What if the problem you showed: x = (1/3)(y^2 + 2)^(3/2), 1

[@marisaatristain1122]

that's what i was just thinking

[@tgw1199]

rotated at x axis = dx.... right? conflict with the 1st problem in this video... 😵

[@tgw1199]

got it... 😅

[@matauka.f.mufungulwa64]

Why are the limits of the curve being changed?

[@RamazChemia]

does that mean that instead of R(x) I can just write F(x) in the formula?

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what if we wanted to calculate the volume sir? kindly share method

[@John-bq1lp]

then u use disks and washers, shells

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[@joppai3518]

Hahahahaha

[@Desm0nd4343]

so basically R(x) is the same as f(x).....or am i wrong?

[@lilangel9691]

R(x) is the radius of the function. It was equal to y because the height of the function, is the function itself. So R(x) = y, and we were given that y is equal to x^3. I hope that makes sense

[@talyamansour]

In 15:15 isnt the derivative of y^2= 2yy'? And not just 2y? Or is this not implicit

[@snide1574]

we are deriving with respect to y

[@tristandiggs406]

Can anyone explain why in the first example why we change the limits of integration? First it's 0 to 2, then it becomes 1-145.

[@nassraljabri1375]

because we integrate by substituation

[@cs-oi4im]

12:57 In question no.3, if we are rotating the curve about x-axis then why would we take radius in terms of y rather than in terms of x?

[@ConceptualCalculus]

He worked the problem as if it were revolved about the y-axes, although he said it was about the x-axis.

[@CapSher]

@@ConceptualCalculus No he didn't. If he was revolving it around the y-axis, the radius would have been equal to the equation in terms of y. Revolving it around the x-axis, makes the radius the equation in terms of x, or just y.

[@trippieblue9207]

Why did you not sub back in for u at 6:10? Can someone please explain? I am very confused about it

[@MikeTheGreatCC12]

It is because he changed the limits. However, if you do not change the limits and leave it as 1 to 2 then you can sub back. I hope that made sense.

[@trippieblue9207]

@@MikeTheGreatCC12 Yeah that makes sense. Thanks

[@bellkranel2698]

@@MikeTheGreatCC12 thank you very much! that makes a lot of sense

[@phlaelogaming7071]

when is f(x) and r(x) different?

[@happycakess]

Wait but why didnt you substitute U back in for the first problem?

[@aslam_9407]

12:37

[@orangenostril]

Why doesn't just adding up the circumferences work? ie: ∫(2π*f(x))dx

[@user-uh8wk5uy9n]

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

[@cho506]

why do we need to change the x values to u values??

[@abiryaakovalmoznino3395]

We’re using a technique called U-substitution

[@MehmetSonerKorucu]

dude in 21:53 if you draw the graphic correct, isn't R(x) going to be 2-x ? because R(x) + x = 2

[@sonyduck8663]

in the first example, the u sub doesn't look correct to me, I'm getting a different answer.

[@huyha9642]

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