Change the order of integration to solve tricky integrals

Рет қаралды 148,862

Күн бұрын

The double integral over a region can be expressed in two different ways. It could be that we write dxdy which means that we integrate with respect to x first and then y (called horizontal strip). Or it could be dydx which means y is integrated first and then x (vertical strips). The big idea is we can convert between these and so can choose whichever is the easier of the two. The steps are:
1) Given an integral, sketch the region implied by the limits of integration
2) Now interpret that region the opposite way, such as vertical strips instead of horizontal strips
3) Write the new integral in the opposite order of integration
4) Hopefully it will now be easier to compute!
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Пікірлер: 113

@angelluisgarciaguzman5598 3 жыл бұрын

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

@shubhamingale7044 2 жыл бұрын

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 Жыл бұрын

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

@ofcanjan 9 ай бұрын

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

@NikhilUpadhyayIITB 8 ай бұрын

IITB? EP?

@NielsLeoLarsen 4 ай бұрын

This man will single handed save my calculus journey.

@Ann-ny1yb 8 ай бұрын

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 3 жыл бұрын

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

@viliusr7928 2 жыл бұрын

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

@heathpatterson3694 8 ай бұрын

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

@luna_lovegood4904 9 ай бұрын

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

@mister_allmond 4 жыл бұрын

Very thorough tutorial!!! Enjoyed watching!

@briandwi2504 Жыл бұрын

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

@diveintoengineering6089 3 жыл бұрын

Thank you! Please keep uploading such wonderful videos!

@joshkr1993 2 жыл бұрын

Nice simple and clear explanation, thanks for the help!

@adresscenter 3 жыл бұрын

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

@mohan153doshi Жыл бұрын

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

@ReidCipriani 10 ай бұрын

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

@mohamedridamahir1960 3 жыл бұрын

love you boss! this is just what i needed

@duartegomes8491 Ай бұрын

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

@Aim4sixmeals 4 жыл бұрын

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

@namhoang5802 2 жыл бұрын

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

@perspicacity89 8 ай бұрын

Fantastic video, thank you.

@videitos09 2 жыл бұрын

You're an excellent teacher.

@user-gz9tb1kc3y 2 ай бұрын

Thank you sir!

@panashepanaxe629 10 ай бұрын

Great work Dr

@hshisnan 17 күн бұрын

Thank you so much sir

@pipertripp 4 жыл бұрын

This was really nice.

@benboy3345 8 ай бұрын

Really helpful for calculus 1 thanks

@isuckatthisgame 7 ай бұрын

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.

@lightningbolt321 2 жыл бұрын

Yoo, I appreciate this!

@rajeshkumarbehera519 10 ай бұрын

Really it's a masterpiece!! ❤

@monicapym8948 3 жыл бұрын

Great video and explanation!! Thank you!!

@DrTrefor 3 жыл бұрын

Glad it helped

@kyletaylor-nelson3888 3 жыл бұрын

Absolute legend :D

@crazyteam9370 2 жыл бұрын

Sir Excellent video! Keep doing such videos 🔥

@esthercinzah 2 жыл бұрын

you are just awesome and genius.

@ojaspatil2094 2 жыл бұрын

Ty!

@architgupta5258 3 жыл бұрын

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

@manuelignacio963 4 жыл бұрын

Could you make one for triple integrals please! Great vids

@emilnystrom7341 2 жыл бұрын

Very nice video!

@mikuculus3720 Жыл бұрын

Great professor

@ayushchoubey635 2 жыл бұрын

Thanks

@zirkq 25 күн бұрын

very helpful

@freshely9309 3 жыл бұрын

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

@dellpi3911 3 жыл бұрын

kzfaq.info/get/bejne/f8eemrtyqKy7ZGw.html

@bilalabbad7954 Жыл бұрын

Good explanation

@continnum_radhe-radhe 2 жыл бұрын

Thank you sir 🙏🙏🙏

@thenellecornejo3724 3 жыл бұрын

MUCH THANKS

@sumankumar_phy0987 3 жыл бұрын

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

@markbell8967 3 жыл бұрын

Thanks again

@shradhzk Жыл бұрын

thanks a lott sir for such wonderful vids

@charliemcgraw5463 3 жыл бұрын

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

@sathyamoorthy8763 2 жыл бұрын

You are a mathematics maniac

@meetkavad1821 Жыл бұрын

helpful 🙏

@AbhishekKumar-jg7gq 3 жыл бұрын

Nice explanation

@hussamalanesi7432 Жыл бұрын

Could you please explain changing of triple integral as well

@farheensultan5559 2 жыл бұрын

Well done 👍

@marahnori9354 Жыл бұрын

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

@whitelightning7580 2 жыл бұрын

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?

@joaomattos9271 Жыл бұрын

Great!!!!!

@suhailawm 2 жыл бұрын

Thanks a Lot Proffesor. im from Srilanka

@JamesLewis2 9 ай бұрын

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).

@frosty8655 3 жыл бұрын

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 3 жыл бұрын

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.

@aashsyed1277 2 жыл бұрын

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.

@ahanshetty9034 Жыл бұрын

Dr. Trefor the GOAT 🐐

@alejandrom8630 6 ай бұрын

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?

@wakeawake2950 4 жыл бұрын

Nice video 👍👍

@hdot2613 3 жыл бұрын

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 3 жыл бұрын

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.

@kunalch.7061 3 жыл бұрын

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

@YassineBoubcheur Жыл бұрын

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

@admirekativhu6783 Жыл бұрын

you are awesome

@somnathsaha7391 2 жыл бұрын

Great

@ismaelrascon99 3 жыл бұрын

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

@DrTrefor 3 жыл бұрын

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 3 жыл бұрын

@@DrTrefor it's probably him😂

@cloroxlysolbleachwhipes8658 Жыл бұрын

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??

@funkysagancat3295 2 жыл бұрын

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

@ronronn3148 3 жыл бұрын

god tier explanation

@dellpi3911 3 жыл бұрын

kzfaq.info/get/bejne/f8eemrtyqKy7ZGw.html

@MariAmmaSar Жыл бұрын

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

@SF-og3fq 3 жыл бұрын

this guys cadence is hard

@user-em8cv4dd6k 6 ай бұрын

why didn't you change the interval after substitution?

@mschecter0078 Жыл бұрын

Dude you are fucking awesome! I finally get it now! I hope I get at least a B in the class so I can stay in the honors program at my university!

@limalshaji6788 3 ай бұрын

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

@obssawega3084 9 ай бұрын

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 3 жыл бұрын

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 3 жыл бұрын

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 3 жыл бұрын

@@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

@SHAHHUSSAIN 4 жыл бұрын

Legend 😊😊💝💝💖💖

@SHAHHUSSAIN 4 жыл бұрын

💝Sir I am big fan of you💝

@abstractnonsense3253 4 жыл бұрын

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

@abstractnonsense3253 4 жыл бұрын

@@DrTrefor Excellent! Thank you in advance.

@maximlavrenko1164 2 жыл бұрын

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

@aashsyed1277 2 жыл бұрын

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

@rajmahajan5158 2 жыл бұрын

Speak is most powerful

@NitinPandey-cv9wi 2 ай бұрын

❤

@Dina-he1uc 3 жыл бұрын

😊😊

@fernandomagallanesjr.9715 2 жыл бұрын

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!

@complexrealmaster5511 4 жыл бұрын

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

@complexrealmaster5511 4 жыл бұрын

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

@wqltr1822 4 ай бұрын

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