Change the order of integration to solve tricky integrals

[@angelluisgarciaguzman5598]

To get the YT algorithm working: Just add Calculus III and the title and tags

[@isuckatthisgame]

The hardest part, I've found, is to know how to plot these functions in order to see where is the region you want to integrate. I need to build geometric intuition.

[@NielsLeoLarsen]

This man will single handed save my calculus journey.

[@shubhamingale7044]

I wasn't understanding how to assign the values of limits and I was worried about it as my exams are approaching. Now I have understand it clearly Thank you Dr. Bazett, you are a life saver. Love from India.

[@rahulkanwar3233]

me too. actually i need more videos on this topic but don't know what to search

[@ofcanjan]

Yes, Indian math professors focus more on solving the questions rather explaining what's happening.

[@NikhilUpadhyayIITB]

IITB? EP?

[@silent_apparel]

Hi India Boy

[@Ann-ny1yb]

This video gives me hope. I was on the verge of breaking into molecules as I did not understand a single word from my Instructor when he explained this. Thank you very much, and now I get it.

[@tomatrix7525]

I’m so glad I stumbled upon this channel, it’s a gold mine. Great work!

[@fatherkeys73]

You explained this so clearly. I’m so happy I found this channel for calc 3

[@duartegomes8491]

I am serious. Thank you, i am an ee student and my professor just taught us how to do this very fast like everyone would know. you're a king, God bless u

[@viliusr7928]

Man you opened my eyes and i finally can understand the logic behind this. That's very great teaching

[@briandwi2504]

Sometimes maths really is like magic! Such a wonderful technique. Thanks for the video. Pure class.

[@Reg4JesusChrist]

One of the best to ever teach it, Sir!!

[@mohan153doshi]

Superb explanation and very succinct too. Great video Dr. Bazett.

[@niranjanarangaraj]

you're saving my calc endsem today thank youuu

[@Aim4sixmeals]

Something needed now that everything is online due to corona until next year

[@luna_lovegood4904]

Omg this saved my life!!!! thankyou so much, i had so much trouble visualizing the inner and outer parts on a diagram and now i understand

[@bonnieamof7033]

Life Makes Sense Now...

[@mananahuja4656]

But when you substituted the denominator equals u , i think you should change the value of limits as well.

[@rajeshkumarbehera519]

Really it's a masterpiece!! ❤

[@monicapym8948]

Great video and explanation!! Thank you!!

[@DrTrefor]

Glad it helped

[@crazyteam9370]

Sir Excellent video! Keep doing such videos 🔥

[@namhoang5802]

great video. All other instruction videos did not go as deep to the basis as this one, I can say for sure.

[@videitos09]

You're an excellent teacher.

[@diveintoengineering6089]

Thank you! Please keep uploading such wonderful videos!

[@ahanshetty9034]

Dr. Trefor the GOAT 🐐

[@mohamedridamahir1960]

love you boss! this is just what i needed

[@charliemcgraw5463]

why did you not change the bounds to match with u = y^4+1?

[@mister_allmond]

Very thorough tutorial!!! Enjoyed watching!

[@ReidCipriani]

Thank you. This concept is simple yet hard to visualize sometimes

[@frosty8655]

can you explain why for dx we use horizontal strips ? My confusion stems from the fact that in single variable calculus when we integrate f(x)dx we view the function as being divided into many vertical strips. I am confused as why in this double integral we use horizontal strips for dx and in single variable integral we used vertical strips.

[@shuewingtam6210]

You may consider the horizontal strip dy being partitioned into several blocks along x-axis. Integrating x first means that these blocks is sumed into the strip dy according to the basic concept of integration. Integrating y then sum up the strip dy into the area contained in the region.

[@hdot2613]

Hi Dr Trefor Bazett. Thanks for the great explanation and solid visuals! I'm a little confused on one part and was hoping you could help clear up my misunderstanding 😁 Why is the upper bound on the inner x integral y^3 and not 8? We have the line x=8 plotted and it's equivalent to 2^3. Why can we not argue that the largest x value of the shaded area is 8 instead of y^3?

[@shuriken188]

Because then you would be integrating the box enclosing the region instead of the region itself. The switched integral is saying "for each y from 0 to 2, add up all the cross-sections of the function at that y where x is between 0 and y^3." Each 2D slice has a different width at each y value. There is one cross-section where the width is 8, and that's when y = 2. But the cross-section at y = 0 has a width (x) of 0. So for y = 0, you have to add 0 area instead of a cross-section 8 wide.

[@whitelightning7580]

After integrating the inside integral, which is a constant 1/y^4+1 how did you plug in y^3 on the top? It is a constant number (1) on the top, why is that one being replaced with the bounds of integration? (y^3) Doesnt it need to be a y in order to be replaced with y^3?

[@JamesLewis2]

It's not obvious, but the factorization of y^4+1 over the reals is (y^2−y√2+1)(y^2+y√2+1); it's the difference of the two squares (y^2+1)^2 and 2y^2. Of course, if you then use partial-fraction decomposition and complete the square and find an antiderivative, you'll end up with a sum of logs and arctans that would be unusually difficult to work with, and *then* you would need to deal with the compositions of those functions with the cube root to calculate the outer integral. To spell it out more, it turns out that an antiderivative of 1/(y^4+1) is sqrt(2)/8*(ln(y^2+sqrt(2)y+1)-ln(y^2-sqrt(2)y+1)+2arctan(sqrt(2)y+1)+2arctan(sqrt(2)y-1)); evaluated at the upper limit, this is sqrt(2)/8*(2pi-2arctan(2sqrt(2)/3)+ln((33+20sqrt(2))/17)), and it turns out that when you evaluate it at the lower limit, the resulting function of x *does* have an elementary antiderivative, but it's far from obvious, and it also is far from obvious that every term cancels except for a couple of terms that add up to ¼ln(17).

[@ismaelrascon99]

Who would dislike lmao. He makes the topic so simple.

[@DrTrefor]

I remember my first hater who went through every single video I had (back when they had like less than a hundred views) and dislike every single one in about 10 minutes:D AFter that I really don't care anymore lol

[@ismaelrascon99]

@@DrTrefor it's probably him😂

[@MariAmmaSar]

You've done your bit to make life on this planet a little less miserable, thank you

[@freshely9309]

Thank you again man, you are saving my career! :D

[@dellpi3911]

kzbin.info/www/bejne/f5LRmIt3eKuKapo

[@dragonbread1790]

Thank you, this was very helpful!

[@esthercinzah]

you are just awesome and genius.

[@continnum_radhe-radhe]

Thank you sir 🙏🙏🙏

[@perspicacity89]

Fantastic video, thank you.

[@suhailawm]

Thanks a Lot Proffesor. im from Srilanka

[@panashepanaxe629]

Great work Dr

[@sathyamoorthy8763]

You are a mathematics maniac

[@joshkr1993]

Nice simple and clear explanation, thanks for the help!

[@benboy3345]

Really helpful for calculus 1 thanks

[@manuelignacio963]

Could you make one for triple integrals please! Great vids

[@architgupta5258]

how to proceed if we are having 2 regions and which to select

[@mikuculus3720]

Great professor

[@obssawega3084]

Doesn't integrating with respect to y, mean that you have horizontal strips? since dy is the width of the strips this means that the strip is horizontal (given that dy is y2-y1)

[@parampalsingh3339]

sorry but i don't understand why integrating with respect to y first would imply the vertical strips and not the horizontal ones...i mean if you see it as as sums of little rectangles (riemannian integral i guess) the dy should imply small rectangles with infinitesimal height and width equal to the value of the function in x...I'm just a bit confused.Thanks for the video though..very informative

[@DrTrefor]

These integrals are always with respect to two variables, just a question of order. I.e dydx or dxdy. Whn doing dydx this is saying first dy then dx. So when doing the first dy you are changing the y values i.e. it is a vertical strip. That strip yes will have a width of dx, but we are imaginging changing it's long height into little segments of height dy and integrating to get the full vertical strip. Then integrated w.r.t dx to add up all the vertical strips.

[@parampalsingh3339]

@@DrTrefor Thanks for the reply...I think i got ..still need to wrap my head around it for a while but i guess i can manage that :D

[@PrinceYadav0702]

Best explanation

[@lightningbolt321]

Yoo, I appreciate this!

[@alejandrom8630]

why do you not replace the upper and lower bounds for dy when doing u-substitution? where if you plug in you would get y =1 and y = 16?

[@YassineBoubcheur]

Is there any analytical algorithm to do this ? Because sometimes bounds may be complicated functions to draw !!

[@adresscenter]

Awesome! Thank you. Please more vidéos of exercices like this one

[@aashsyed1277]

I have a question. Let's say I wanna solve Integral from 0 to 1 of the integral from 0 to √(x²+y²) of 1/y⁵+1 dy dx . How can I solve this integral? If I wanna change the order, but I have a two variable function in the inside integral. So can I use polar coordinates in this integral? It looks it's gonna go bad.

[@ronronn3148]

god tier explanation

[@dellpi3911]

kzbin.info/www/bejne/f5LRmIt3eKuKapo

[@pipertripp]

This was really nice.

[@Bold_new_city_of_the_south]

I've been doing so much solving of partial differential equations and nonlinear odes.. I forgot about all of this.

[@hussamalanesi7432]

Could you please explain changing of triple integral as well

[@bilalabbad7954]

Good explanation

[@fernandomagallanesjr.9715]

AMEN FOR YOUR VIDEO I GOT STUCK ON A COSX/X when we didnt even discuss yet Taylor series. finally got a way to solve my problem by doing the changing of order. All this through your help. Tysm!

[@shradhzk]

thanks a lott sir for such wonderful vids

[@kyletaylor-nelson3888]

Absolute legend :D

[@cloroxlysolbleachwhipes8658]

my question is how am i supposed to know what the graph looks like on an exam (especially sqrt(x). is there a mathematical way to figure the new bounds for integration??

[@KaleabGetachew-b2i]

why didn't you change the interval after substitution?

[@kunalch.7061]

Nyc explanation... But try to stretch the length of the video by increasing quantity of videos as well as Difficulty level ! Thnks.

[@ojaspatil2094]

Ty!

[@maximlavrenko1164]

Hmm, I don't get what this integral represents... There is no x in the function, so what volume is this?

[@marahnori9354]

Doctor, what if the root of x is a substitution of the cube root of x, what would the solution be?

[@SF-og3fq]

this guys cadence is hard

[@AbhishekKumar-jg7gq]

Nice explanation

[@limalshaji6788]

an actual g i never comment bruh but ur a life saver holy shit

[@meetkavad1821]

helpful 🙏

[@farheensultan5559]

Well done 👍

[@sumankumar_phy0987]

How would you calculate the center of a force field, is there any method?

[@ayushchoubey635]

Thanks

[@emilnystrom7341]

Very nice video!

[@SHAHHUSSAIN]

Legend 😊😊💝💝💖💖

[@SHAHHUSSAIN]

💝Sir I am big fan of you💝

[@hshisnan]

Thank you so much sir

[@孫傅康]

Thank you sir!

[@markbell8967]

Thanks again

[@funkysagancat3295]

sometimes the function is really weird and I can't draw it, any tips?

[@abstractnonsense3253]

Hi Trefor. Do you intend to make videos on the vector calculus section of Calculus III and the V.C. integral theorems? Thanks!

[@abstractnonsense3253]

@@DrTrefor Excellent! Thank you in advance.

[@wakeawake2950]

Nice video 👍👍

[@zirkq]

very helpful

[@thenellecornejo3724]

MUCH THANKS

[@scuti7073]

Bros so goated

[@Thehumee12]

Dude have naturally 2x speed

[@rajmahajan5158]

Speak is most powerful

[@nagabrahmeswarpusapati6930]

thankk you professor

[@zish3655]

I was so ready to skip the chapter before I came across this video

[@ΚωνσταντίνοςΛαζαρίδης-ξ9ι]

thanks!

[@wqltr1822]

Int(1/(1+y^4))=Int(y^-2/(y^-2 + y^2))=1/2Int((1+y^-2)/((y-y^-1)^2 +2)-1/2Int((1-y^-2)/((y+y^-1)^2 +2)) Some nice subs appear, which eventually lead to a solution... Yeah maybe switching order first is nicer lmao

[@aashsyed1277]

It is possible to solve the integral of 1/x⁴+1 dx!

[@admirekativhu6783]

you are awesome

[@pinsaramallawarachchi8201]

You are doing a social service

[@somnathsaha7391]

Great

[@joaomattos9271]

Great!!!!!

[@Bold_new_city_of_the_south]

I forgot a lot!

[@complexrealmaster5511]

Can you remake a video about strong induction? The video has low quality. Anyway, I liked the video.

[@complexrealmaster5511]

Trefor Bazett Thank you. Keep doing what you are doing.👍