Converting double integrals to polar coordinates (KristaKingMath)

► My Multiple Integrals course: www.kristaking...
Learn how to convert double integrals from cartesian coordinates to polar coordinates.
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If you could use some extra help with your math class, then check out Krista’s website // www.kristakingm...
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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)
Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”
So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student-from basic middle school classes to advanced college calculus-figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here: www.kristakingm...
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[@amardubb8834]

I like how you don't skip any steps. I learn really well because of that! THANKS AGAIN!

[@AGNIVO-kf2bl]

Can you please specify why dx.dy gets replaced by rdrd(theta) ?

[@simkachik]

@@AGNIVO-kf2bl dude, it's easy, not even worth mentioning, just do as they say and don't think too much))

[@My_NameJeff]

I have my year-end exam in 2 days and i've been trying to understand this thing for the last 2 months(i am very weak at calculus)! i am so happy that i reached this place and was able to learn this thing in just 12 minutes! thank you so much,you've just increased my hopes of saving the semester! :)

[@kristakingmath]

vatsalya singh I'm so glad to hear that! Good luck on your exam, I hope it goes great!

[@My_NameJeff]

:)

[@HackingDutchman]

Great video, very clear instructions! I like that every single step gets explained, even those well known little ones, so you can't overlook anything.

[@drewsteps]

Your explanation is clear, thorough and succinct. I've never seen that before for math videos. Please post more videos.

[@militantpacifist4087]

That’s literally one of the problems I have on my homework. 😂

[@riteshrk18]

Wow you have solved it and nailed bro😎😎

[@mariorafaelbritopavon456]

yooooo same 3 years later

[@bilalrahim2076]

@@mariorafaelbritopavon456 yooooo same 4 weeks later xD

[@dez7800]

@@bilalrahim2076 same lol It's in the stewarts manual which is used worldwide sooo.... Makes sense !

[@allanhallsolorio2593]

X2

[@mrunalshah3660]

I understood each and every concept behind the question. Thank you ma'am to explain with such a great method of teaching. 😀😀😀

[@r.f2173]

thanks, this helped. My Calc 2 prof last year skipped polar coordinates altogether so now I'm figuring this out on my own

[@marcibelasky2862]

If using the u substitution don't we have to change our interval? So in our case we would have to put our intervals (0,2) into u and we would get a new interval (0,4) with which we continue to calculate.

[@staydank420]

Great video!! One thing I noticed though: when you u-substitute for r^2, you didn't find new bounds of integration in terms of u and therefore wrote a false statement by leaving the bounds in terms of r. You rectified your mistake in the next line but my calculus professor will mark points off for that on an exam and I am hyper-vigilant of that mistake. He said to, in the least, leave it as an indefinite integral until you back-substitute for u.

[@h3nnn4n]

I noticed that too.

[@ashutosharya6221]

Derrek Schmitz , you are right after substitution, we have to make changes in limits according to the substitution.

[@robconnell5549]

Yes. The dud(theta) upper integral limits(just above the centre of the screen view) of the right integral sign should be 4. Krista corrects it on the next line when she goes back to using 'r' and using 2 for the upper integral. Now hopefully this will help solving an integral equation using the angle 'theta' to get a 2d solution of a second order random field. We will see....

[@acealexander6852]

I've always re-subbed the U value back in and then evaluated for the variable in this case r. it works both ways.

[@riteshpatidar9184]

Yea..

[@assassinjohn]

best teacher who saved my semester :)

[@giulpiano]

legit tho

[@yohaneilunga4404]

Good maths steps am really behind on how to find the boundary region and on how to get the limits of a double integration needs your help

[@caelawnn]

OH MY GAW... KRISTA... THANK YOU. I WAS STUCK ON THIS FOR SO LONG BEFORE I STUMBLED UPON YOUR VIDEO!!!

[@kristakingmath]

:D

[@greenlizardballs]

lovely explanation! id look at other videos and go "wait but where did you.. but how did you.. where did that.. " you don't get that here! :D

[@kristakingmath]

+greenlizardballs It makes me so happy to hear that! :D

[@superdupe8]

this video makes me want to burn my calculus book

[@shane2678]

Excellent video. 5 *'s. You really helped me understand integration where you've to convert co-ordinates from Cartesian -> Polar. Thank you so much. The folks crying about the limits being 4 rather than 2, cause u=r^2 may be correct, but they're just being pedantic imo. You still got the right answer!

[@justinnussbaum6550]

Just a bit of advice to make this a little easier. Rather than integrating between -Pi/2 to Pi/2, you can integrate from 0 to Pi. It will make evaluating the integral much easier and will result in the same answer. If you are shifting the limits, there is the same distance between those points. If you can make 0 as your lower limit, evaluating your integral will be much easier and less of a chance to make a mistake.

[@uhuihiuihiuh3042]

You can also go from 0 to pi/2 and multiply by 2 before evaluating the integral so you get the proper area. The 2 and the -1/2 cancel out which leaves a - hanging out front. This way the only fraction you deal with is the pi/2 and like you said, using 0 as the lower bound simplifies the work load a lot.

[@electronicwoe]

I seriously appreciate all of your videos. Thank you. :)

[@kristakingmath]

You're welcome!

[@TheCommander454]

Pete Dietl me too

[@NikhilKumar-sl5ep]

i have learned a lot from you. thank you so much for sharing this lecture!

[@moinaction]

Nobody, I mean nobody explains calculus better than you do. Ignore those thumb down those are people who can't explain closer to what you do. We appreciate you. Thanks

[@Spark-br9ut]

I know this is years after you made this video, but thank you so much! You are a life-saver!!!

[@kristakingmath]

You're welcome, Spark, I'm so glad I was able to help! :D

[@carlellis9647]

You did a great job explaining double integrals to polar coordinates in this video. I went from having very little knowledge in this subject to being able to solve problems. Thanks so much!

[@kristakingmath]

+Carl Ellis So glad I could help!

[@abeekuampah2419]

How am i just discovering this channel???????????? You're the BEST. Thanks!!!

[@shitpostheavy]

Great video thanks. One question though: because you did u-substitution, do you not have to change your upper/lower bounds for du? In this example, u=2r so therefore the bounds should be u=2(0) = 0 for lower and u=2(2)=4 for higher? Or is it a rule that when you sub back in r^2 for u, the bounds should remain the same? (I hope I am making some kind of sense).

[@kristakingmath]

You're making perfect sense, I know exactly what you're talking about. Technically, you should change the limits of integration when you make a substitution. However, you only need to change them if you plan to evaluate over the interval WITHOUT back-substituting. If you back-substitute, then you'd just have to change the limits of integration back to what they were originally. Since I always back-substitute, I don't worry about changing them. I hope that helps clarify!! :D

[@rachelnanshija251]

Thanks for this. My book has iterated integrals in polar, cylindrical and spherical coordinates all packed into one chapter and so the author just glances over all three. So yea, thanks again, and thanks for being so thorough. You saved me a lot of headache and frustration.

[@kristakingmath]

+Rachel Nanshija Thanks for the comment. Glad I could help!

[@kristakingmath]

Great timing! :)

[@daherahmed2817]

i love you really .. math without you like coffee without sugar .. :) thanks and keep going

[@chenkang8811]

Can't live without Krista during coronavirus thank u so much

[@kristakingmath]

You're welcome, Chen! Hope you and your family are safe and healthy! :)

[@punditgi]

Krista is the Queen of Math!

[@manedurphy]

You are literally my favorite. All your videos are really clear and I actually learn :)

[@ElementMucho]

You’re the best calc explainer yet

[@thabi.b7128]

Your explanations are really clear and your though process/method is beautiful... Thank you…

[@kristakingmath]

Mthabisi Bokete You're welcome, I'm glad you like the videos!

[@farzanasultanasoha5350]

just great. i watched a lot of video on this topic but this one just made my whole concept clear. thank u so much.

[@inteusproductions]

Been watching your vids for a year now, they are always so clear, thanks so much dear :)

[@kristakingmath]

inteusproductions You're welcome, glad they've been helping!

[@SGF08Y]

You explain things much more succinctly than any of my past teachers/lecturers :)

[@kristakingmath]

Thank you very much!

[@footballistaedit25]

This is the best explanation. You explain the concept clearly. Thank you so much

[@congshen]

I can only concluded that you're better than my further calculus lecturer!! Thanks for your detailed explanation!!

[@kristakingmath]

you're welcome, i'm so glad it helped!!

[@JoseCamposCR]

This is so much better than the way my teacher explained it, or at least tried to.... Thanks!

[@kristakingmath]

You're welcome, glad it helped!

[@kristakingmath]

yes, this is usually calc 3 stuff. :)

[@mkusasakala3861]

last semester i missed a lot of calculus classes but with the help of your vids together with other guys vids on youtube i passed my calculus exam as though it was some kind of test on general knowledge.Now look at me, i am missing classes again this semester :p

[@kristakingmath]

LOL, well I'm so glad the videos helped!!

[@Dawlada]

You are really amazing. Thanks for providing free lectures.

[@kristakingmath]

My pleasure, Dawlad! I'm happy I'm able to help! :D

[@tc8257]

Thanks! I'm a big fan of your teaching style.

[@chimbinikangwa2940]

Your tutorial is still live here, thanks for the wonderful explanation you got my finger

[@11ans22]

Thank you mam it is very easy to learn from your videos rather than from class

[@kristakingmath]

You're welcome!

[@TheISP]

Thank you very much for this beautiful explanation! It's helped me a ton in understanding double integrals!!! 😍

[@kristakingmath]

Glad it was helpful, Ishaan!

[@mdfaizan1937]

i always get stuck while solving this problem, but i had almost mastered it. Once again THANK YOU very much.

[@FreakfestTV]

She’s the best teacher ever

[@MFDan69]

so so so happy i found this!!! LIFE SAVING

[@kristakingmath]

+Cesar Solorzano I'm so glad it could help!

[@ironman5034]

Just a few hours before my exam, you are a life saver🙏

[@kristakingmath]

I hope the exam went great! :D

[@lyradmuk511]

The best explanation I’ve seen so far. Superb! Wish you were my lecturer

[@kristakingmath]

I'm so glad you liked the video! :D

[@ruffinist]

seriously you're like khanacademy 2.0, thank you for the videos you're a life saver.

[@kristakingmath]

I'm glad they're helping! :)

[@ChiDeli]

SERIOUSLY. You have been leading me to the promise land this semester, thank you so much.

[@Farrukhw]

Thanks a bunch Krista for such awesome an explanation. Really helps, especially in evaluating the limits, a point where I am always confused...

[@kristakingmath]

You're welcome, Farrukh! I'm so glad it helped! :D

[@hellonoone2014]

Great video! I was absent in class due to sickness when this was taught and so I had a difficult time catching up. But this really helped a lot. Thank you! You just gained one more subscriber :D

[@HaCaralho]

Thanks! you have no idea how much trouble i had to understand this before.

[@kristakingmath]

You're welcome! I'm so glad it helped!

[@RakeshChaudhary0908]

Hi @krista - Is the final answer correct ? Actually, the limit of "u" was not included in the integral. You continued the integral calculations with the limits of r only. limits of u will be 0 to 4

[@hao2000ki]

she immediately plugged r^2 back into u, so you dont need to convert the bounds to u

[@pholarised6911]

true

[@jyaswanth9688]

Ya It's true

[@geezerlb4066]

but why does dxdy become drdø multiplied with r?

[@KyleBroder]

I cover that in my video on double integrals in polar coordinates.

[@thanhtantruongvinh99]

It's the determinant of the Jacobian.

[@iiamtrista]

it the 'jacobian' which is just a scaling factor when you change from the cartersian cord. to polar coordinate

[@TheFarmanimalfriend]

so when you integrate r you get r^2/2 Surfaces are length measurements, (r), squared.

[@raghurrai]

Great video ! Thanx for taking the time out for this. :)

[@devinpatin552]

I am no longer lost, thank you so much!!

[@kristakingmath]

Glad it could help!

[@Drevinum]

Wow read some stuff about this on some other sites and understood nothing then I watched this 13 min long video and now understand everything THANKS!

[@kristakingmath]

Love that! I'm so glad it could help! :D

[@asemnafiz]

What a coincidence! I was stuck at this problem when I decided to check your video for help.

[@PrestonTwentyTen]

Ohhhh, the good ole multi calc days. Probably the best calculus there is!!!

[@satishnair5630]

7 years ago when i was in 6th grade, this video was uploaded i had no clue what calculus meant. fast forward 7 years , still i have no clue and thats why iam here . thanks for uploading this video.

[@kristakingmath]

Lol, you bet! I just hope the video helped! :D

[@lakshminarayanareddypadart9807]

thanks it was very helpful.actually i saw only one video and could solve the rest. ur explanations are too good .well thanks again.

[@qhairilsalim]

wish me luck for my exam tomorrow. my lecturer taught this for 2 weeks but i understand your 12 minutes video better. thanks

[@kristakingmath]

I'm so glad the video helped, and I hope the exam went great! :D

[@sumitnayak22741]

Good i have my doubts in multiple integral but after watching this video i clear my all confusion thank you and watching from nepal

[@kristakingmath]

I'm so glad the video helped, Sumit! :)

[@kristakingmath]

Glad I could help! :)

[@fatiswealth7865]

probably the best video out there of explaining this example!!!

[@kristakingmath]

Thanks! :)

[@mavikaktus5712]

Oh thank you! You saved my time and life. I am grateful to you

[@alokkumar8793]

Thank you big sister , my doubts were regarding ,that how to take theta, which is cleared in one example , thank you for your help🙏🙏

[@Augustine_354]

Great video! But at 8:52 in the second integral the upper limit should be 4 :)

[@Rohitsingh-zd2qi]

Piotr Rudnicki it should be root(2)

[@antonyprajin895]

ok i thought it was just me thank you random commenter from 8 years ago

[@LightSpinAngler]

For simplicity I would integrate the theta first and leave it outside as a constant, since you are always going to get a constant for theta with these types of problems. It eliminates some of the mess of the integral and you will arrive at the same exact answer without extra Algebra.

[@frankcarey250]

Now I am feeling confident to evaluate all this type of integrals. @9:32 Limits should be changed for 'u'. Great explanation! And btw, you have lovely voice 😊

[@lauramunoz4986]

You are the absolute best. You and Sal and JustMathTutoring are the best KZbin teachers. Thank you guys, you are appreciated!

[@kristakingmath]

Aw, thanks! That's great company you listed me with... I'm so glad I can help! :)

[@karolyfodor1391]

Thank you very much, your explanations are helping me maintain an A in calc 3.

[@kristakingmath]

That's so awesome! Thanks for letting me know! :)

[@raullopez88]

You're the only reason I passed my class.

[@kristakingmath]

I'm honored, and I'm so glad you passed! :D

[@mikeahmed7092]

Excellent explanation! Really thankful for the clear example.

[@kristakingmath]

Thanks! I'm glad it could help!

[@shantanugolui1602]

Hey thanks mam for giving me a such good explanation on this topic it really helps me for understanding this topic well

[@kristakingmath]

Oh good! I'm so glad it helped!

[@jayt7295]

You forgot to change the intervals after substituting u, it should be 0 to sqrt2 at 08:47. Please address this problem if I am wrong.

[@saadanwar4937]

My prof gave these problems for hw without actually going over them in class. You save my life. As if school is life.>.> Nope .......but thanks a lot.

[@kristakingmath]

+saad anwar So glad I could help!

[@real01s.50]

Excellent, saved me so much time for my exam studying !

[@kristakingmath]

I hope you rock the exam! :D

[@jansugalski546]

Thanks. I've been trying to understand Maxwell's equations and the vids on double integrals has been helpful

[@lawalmichael8848]

You are simply the best! Thank you!

[@mariarobson5803]

really I just understood everything .... final day before my exams

[@kristakingmath]

+maria robson I'm so glad it's making sense now, good luck on your exams!

[@alirouifed1586]

same here XD

[@impor200]

For the angle in this question is from 0 to π\ 2 because it says the area is confined between the function x and y axis

[@vigneshadithyan]

this video was too helpful for me thank you for publishing this video

[@kristakingmath]

+Vignesh Adithyan Glad I could help!

[@mohammedwajid5481]

Thank you maam...U did really saved me with the lesson. .... Explained very well

[@kristakingmath]

You're welcome, mohammed! I'm happy to help! :)

[@adamg3840]

These videos got me through university! Thank you so much

[@IL0VETHESUN]

Really helpful. Double integrals were amazingly easy until we got to this point, thankfully now they're easy again.

[@kristakingmath]

Glad I could help! :)

[@newyorkguy158]

super clear. easy to understand every step. thanks.

[@morislubin4502]

Nice and clear explanation!

[@kristakingmath]

Thanks!

[@AmitKumar-uj2sb]

very well explained madam..calculus with your sweet voice..its like icing on cake..;-)

[@kristakingmath]

+Amit Kumar Thank you very much!

[@jackrotter9434]

This was an example we did in class, and it was very rushed and done in half the time. You go at a perfect pace and I understand it so much better now. Can you be my professor?

[@kristakingmath]

I'm so glad the video made sense and helped clear things up!! :D

[@aprameypoojar6012]

this video made concepts more clear..

[@kristakingmath]

+Apramey Poojar Great! I'm so glad it helped!

[@360xi]

You make it look so easy!!! Thank you so much for your videos!!!!

[@josht7238]

thank you so much you're such a great teacher

[@alext90210]

That was awesome, Thanks! Helped me understand stuff for my exam :)

[@kristakingmath]

So glad I could help! :D

[@gojibearies]

Wow, that was immensely thorough. Thank you!

[@kristakingmath]

You're welcome!

[@courtneygreer961]

Much better than my teacher! Thanks soo so much!

[@trezzix8079]

Wonderful explanation, thanks!

[@gormeliqyan4031]

Very clear explanation. Thanks

[@kristakingmath]

You're welcome, I'm glad it made sense! :)

[@vkimchi628]

6:22 we don't replace dtheta with dx while converting into polar coordinates...the thing we always look for when deciding the limits of dtheta is the limits of y and x ...and then we decide the limits ..is it so? If the limits of y are from +ve to -ve y axis and the limits of x are from +ve to -ve axis..then the theta limits will be 0 to 2pie..am I right?

[@godfreddy5844]

I LOVE YOU LOVE YOU LOVE YOU LOVE YOU, U JUST SAVED MY ASS BIG TIME xDDD i have final exam tomorrow and i didn't understand anything about this subject, and was about to give up after watching every video possible, then i saw this video and thought to give it 1 last try before i drop calculus3, AND I'M SO GLAD I WATCHED IT, THANK U

[@kristakingmath]

+Hosam Hassan Awww thank you so much! I'm so glad that the video helped and that you decided not to drop the class. Good luck on your final, I hope it goes great!!

[@godfreddy5844]

+Krista King | CalculusExpert.com thank u so much :)