What does a double integral represent?

► My Multiple Integrals course: www.kristaking...
It can be difficult to visualize what a double integral represents, which is why in this video we’ll be answering the question, “What am I finding when I evaluate a double integral?”
In order to answer this question, we’ll compare the double integral to a single integral, so that we understand exactly how to transition from single variable calculus into multivariable calculus. Every piece of the single integral, like the integral, the bounds or limits of integration, the function which is the integrand, and the differential (usually dx) will all translate into a corresponding piece of the double integral.
If we want to describe these with words, we can say that for the single integral, we’re integrating a single variable function f(x) over the interval [a,b], using vertical slices of area, in order to find the total area under the curve f(x) but above the x-axis.
In contrast, we can say that for the double integral, we’re integrating a multivariable function f(x,y) over the region R which is defined for x on the interval [a,b] and for y on the interval [c,d], using vertical slices of volume, in order to find the total volume under the surface f(x,y) but above the xy-plane.
Skip to section:
For the single integral:
0:36 // Sketching a single variable integral
6:32 // Building the area equation for a single integral
7:10 // Moving from the geometric estimation to summation notation
9:24 // Moving from summation notation to the single integral
For the double integral:
10:39 // Sketching a multivariable double integral
20:38 // Building the volume equation for a double integral
21:16 // Moving from the geometric estimation to summation notation
23:22 // Moving from summation notation to the double integral
24:25 // Summary
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● ● ● CONNECT WITH KRISTA ● ● ●
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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[@kristakingmath]

Hello again! I'm excited for the new school year! I'll be posting again regularly, starting this week, with how to interpret a double integral. :)

[@user-mq4qj2bl7g]

Krista King good job 👏🏻 😘

[@ShoaibAmjad]

Thanks!

[@hulkhoganspssy]

thank you for this, my calc 3 class is taking off from here, so this is super useful

[@taylor-mc2vc]

Krista King nice topi plz express triple intrigal

[@kristakingmath]

Awesome, I'm so glad! Good luck in Calc 3 this semester! :D

[@1973jdmc]

You are an ANGEL sent from MATHS HEAVEN to help us poor souls to see the MATH LIGHT- AND what a brilliant job you are doing- thank you from one of your many disciples....

[@isaact.477]

I like that you delved more into the concept and visual/geometric interpretation of a double integral. This was a wonderful, conceptual video, and I wish you did the same thing for all your other ones. Well done!

[@andrewkoulogeorge2569]

the best part about youtube vids like this is people gain value years after they are made. thank you !

[@kristakingmath]

I agree, Andrew, it's awesome! So glad this helped! :D

[@fungaimandikuwaza3374]

If we had many Kristas around the world who approaches math concepts in the manner you do, surely many students would find Math enjoyable and fun. Excellent work Krista and thank you for sharing such an important piece of work!

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Glad that I watched this video. I took calculus some 28 years ago and only recall the basic concepts of limit and Riemann sums. So amazing that such a simple concept is so POWERFUL

[@123goforward]

You are the best. You make everything so easy to understand and yet preserve the beauty of the subject matter.

[@kristakingmath]

😊

[@nathanzbrazil]

You earned my like the moment you drew the stick figure looking at the rectangles. Little details like that are so helpful in understanding these concepts. Thank you for explaining so well!

[@kristakingmath]

I'm so glad it helped!! :D

[@snir143]

You've opened my eyes, literally. Thank you Krista :)

[@blackwolfkills]

Great understanding+great concepts+great explaination+calm voice=great video.

[@mike2884]

One of my NEW fav channels! Thank u!

[@notSavant]

Best explanation on yt by far... Great job Krista!

[@billyobrien3279]

i cant believe how awesome this is, so clearly and simply explained. very nice!

[@charity4613]

This is the best explanation of an integral I have seen on KZbin

[@superawesomeboy]

Not only did this explain Double integrals perfectly, it also summarized single variable integrals very nicely. Wish I had found your videos earlier!

[@cykoshadow]

Please come back Krista I need you to be my Calc 3 professor

[@kangmoabel]

Krista king and organic chemistry tutor are the best calculus teachers in the world ❤❤❤

[@brandonteller5360]

I know a million people have probably thanked you since you posted this video but, I needed to thank you because this helped a lot! Thank you!

[@kristakingmath]

You're welcome, Brandon, I'm so glad this helped!! :D

[@SmileEvErydAy]

Why this video is so underrated. Never see an explaination before. Thank you

[@kristakingmath]

I'm so glad you liked it! :)

[@rocketmanfossel1174]

you are a savior. Bow down to the math goddess 🙇‍♂🙇‍♂

[@arundey6673]

I AM A 0 PERSON IN MATH, BUT YOUR EASY EXPLANATION ENTERTAIN ME REALLY AND TEMPT ME TO DELVE INTO MATH. THANK YOU MY BELOVED SISTER TEACHER. Arun dey

[@rupayanjana6852]

wow! I was not understanding the significance at all but ur pictoral explanations clear all my doubts.Thank u so much

[@age_a587]

Now this is what I was looking for. Feeling short of words to express my gratitude towards you. Excellent video!!!

Amazing work, i understood every single detail :)

[@kristakingmath]

Awesome!! :D

[@johnholme783]

This video gives you a good intuitive grasp of double integrals! Thamk you!

[@ahmadyahiaabuhajar6778]

10X a lot! This made it completely clear for me. I didn't even know that double integral represented a volume, I guessed it'd represent an intersected area between two functions, one of them is Y(x) and the second is X(y).! Keep it up🤞

[@kristakingmath]

So glad it helped, Ahmad!

[@tomspoors768]

Hi Krista, this is a beautifully sustained 25 minutes (and a bit) of clear exposition. Thank you!

[@EriiikaGuerra]

Best overview in double integrals ever! And your voice is so soothing.

[@kristakingmath]

Thanks, Erika! So glad you liked the video. :)

[@Shadow2309Child]

Best presentation I've seen. Simple enough and very understandable. Keep up the great work!

[@kristakingmath]

😊

[@rajan9064]

subscribed as soon as I heard your sweet voice, very well explained, you are an amazing teacher

[@kristakingmath]

Thank you so much, Blackfyre, I appreciate the sub! :D

[@dudleydooright]

If this lady had been my math teacher, I'd be a mathematician today.

[@theodoresweger4948]

You are right I got straight A's in calculous, and now am doing rewiews, the integretion process is easy for me, but still hanve trouble visualizing double and tripple boundaries!!!

[@joliettraveler]

This is fantastic. Now I know where n and m come from and also R and A. Many many thanks. I have a much greater understanding this topic. I can't thank you enough

[@kristakingmath]

You're very welcome, joliet, I'm so glad this helped clarify! :D

[@nicklevelsilva6746]

I like it very much, It was very difficult to me to understand by the books, but your video just simplify everything, thank you!

[@kristakingmath]

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

[@srimouryadanthala8699]

The way you explained was Extra-Ordinary !! ,Thanks for helping in understanding the double integral !!!.

[@saurabhraghuvanshi4262]

Beautifully Explained...! Thank you for the detailed illustration of double integral.

[@kristakingmath]

You're welcome, Saurabh, I'm so glad you enjoyed it!

[@AlePreludioFinal]

Thank you, Krista, the explanation has a good stepping-difficulty. And colour-code is awesome too!

[@reg-net8956]

Love your vids Krista; you and Rob Leonard are the best.

[@owen7185]

Thank you so much Krista, your videos are great and so helpful. You were born for this

[@uswazikwetu5918]

Thank you very much for the wonderful presentation. Very insightful.

[@fquiroz6592]

I was supposed to see this on this semester but my professor decided to leave it out of the program. So thank you! This will be useful.

[@kristakingmath]

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

[@ivornworrell]

*Thx Krista King for this vid; your explanation is pellucid and you are indeed erudite in mathematics.*

[@kristakingmath]

Thank you, Ivor! :D

[@kina4288]

Very detail but concise at the same time. Would live to be your student.

[@chrisdiner7170]

Crystal clear amazing video. Very detailed.

[@055nirupamkolanki8]

Awesome.....now it makes sense why i should study more about them and i will.....thanks krista🖐️🖐️🖐️

[@roaahedaya1779]

Your explanation is very simple and precise.Thank you for your lesson 😍😍

[@kristakingmath]

You're welcome, I'm glad you liked it! :D

[@jee9maths]

Simply superb...Awesome...Please upload this kind of videos more ... It will be so much helpful to Mathematics lovers...Thank you so much madam...

[@cohallel.7]

I love the way you present! Your "SIMPLICITY" ways with your golden and sweet voice makes it lovely and interesting! Your explanations are more crucial and impressive than a full day stupid lecturer. Thank you Krista again and again for devotion and dedication! So are sweet in and out! Cause you make the bitter and difficult part sweet! That flows in a lovely manner!

[@mjs28s]

very well done. +1 for sure.

[@kristakingmath]

Thank you so much! :)

[@ghanshyambharate2130]

thanks , now i really understood what is mean by double integration.

[@hernandezreyescarlosdaniel8824]

You are an amazing teacher, thank you!

[@kristakingmath]

Thank you so much, Carlos! :D

[@alianjohnson6035]

that's fantastic - now what applications would that have in the real world other than finding the volume as you have demonstrated.

[@khushalsahni6058]

You have a beautiful voice, thanks for the video 😊

[@ther7554]

I actually laughing on my concepts now wht I assumed it to be and wht it is ...the difference between them is like difference between sky and ground .,.but this never become obstacle in solving calculus problems as all it was just same mechanical process but today after knowing basics my soul got peace

[@ketkisonawane5048]

Can't thank you enough! You are an amazing teacher!!! Kudos to you! 😊

[@kristakingmath]

Thank you so much, Ketki! :D

[@apderic1079]

I love your math lessons

[@Wichamp59]

Great Video Krista! I wish you would have mentioned that da = dxdy just to show how the dx and the da relate. Could you do a video on what triple or more integrals represent? I have always been confused what those meant since a double was already volume. Thanks!

[@kristakingmath]

I'll try to make one! And great point about the dA=dy dx... I appreciate the feedback! :D

[@duhirwejeannepomscene8635]

You are indeed Krista King!

[@nzsystems]

Thanks Krista! This was another great video.

[@Kamnuma]

Thank you Krista!

[@kristakingmath]

You're welcome, Sofia, thank you for watching! :D

[@harshshrivastava1362]

Awesome I was stuck on this topic for like a week Thank you 😊

[@kristakingmath]

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

[@dsp6742]

Thank you so much, Krista. I now have a different perspective towards multi-variable calculus.

[@kristakingmath]

You're welcome, Dev! I'm so glad you liked it! :)

[@migo77]

Thanks for the nice illustration!. Could you please share which tablet are you using here & which white board software?

[@SeekTheTruthAndTruth]

You made it easy for me. thanks KRISTA .

[@your-pretty-teacher]

I just say that I love you . Excellent ob .1st time some 1 clear the most difficult concept.

[@ayushpatel582]

It's really Wonderful video😀... Put some more video like this about fundamental I loved it 😍 There are so many excellent and wonderful teachers here in India but the problem is that all are not on social media or KZbin.

[@kristakingmath]

I'm glad you liked it, I'll definitely be making more videos! :D

[@EmilioJosedeMatos]

I love your videos Krista! You are great. Your videos have been helping me since I was studying Calc I.

[@kristakingmath]

Wow, awesome! I'm so honored, thank you so much for your support! :D

[@newjade6075]

❤thanks a bunch

[@deepakbellur9676]

A very cogently presented video. But I would like to know, just as in the case of a velocity-time graph, the area under the curve represents the distance travelled, what various things could the volumes under surfaces given by double integrals represent? Thanks once again for the excellent video.

[@0503696580]

what a brilliant way to teach, please keep going

[@kristakingmath]

I will, thank you so much! :D

[@shivamtiwari3679]

Nice explaination. I got cleared my concept,thru urr video. Thanx a lot.

[@kristakingmath]

You're welcome, Shivam, I'm so glad the video cleared things up! :)

[@TinyMaths]

Now, THIS, is how you introduce a subject; just wow.

[@rahulkumarsingh153]

thank a lot...

[@lmorris6170]

Outstanding!!

[@anupkumar9232]

Brilliant work.

[@kristakingmath]

Thanks, Anup!

[@vishalsharma9417]

very satisfying video.thank you

[@kristakingmath]

Thanks, Vishal, so glad you liked it! :D

[@kyrpangdkhar6985]

Thanks you mum........ Love you... God bless you

[@pablobribiesca2596]

A double integral represents area...if you are integrating that function over a function of x and y, it gives the volume, because you are stacking multiple areas together. Just like a single integral gives length. Yet when integrating over a function of x, you are adding multiple lines together which gives area. Hope this clarifies things.

[@aniev.t6597]

Thank you somuch Mam..A great job Mam,The method of teaching is highly impressive.I would like to know the technical packup

[@kristakingmath]

Thank you, Anie, I'm so glad you liked it! :)

[@thatskap]

I'm just confused on why there needs to be two Sigma signs for the double summation. The way you wrote it suggests that there will be points: (x1,y1) (x1,y2) (x1,y3) .... (x1,ym) followed by (x2,y1) (x2,y2) (x2,y3) .... (x2,ym) (x3,y1) (x3,y2) (x3,y3) .... (x3,ym) ... (xn,y1) (xn,y2) (xn,y3) .... (xn,ym) Instead, you showcased pairs of points (x1,y1) (x2,y2) (x3,y3) I'm having trouble understanding the notation and I'm wondering why you wouldn't write it with one Sigma sign that such that it is: n lim as n -> infinity Σ f(xi,yi) * ΔA i=0

[@SidK26]

Thank you Krista...I was wracking my head.. .it was a great relief!

[@kristakingmath]

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

[@chrischauhan1649]

Thank you very very very much, I understand this perfectly.

[@kristakingmath]

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

[@jumpropecommunity6880]

Yeah.. Its really helpful.. Video is quite bit little lenthy but a very nice explanation has been given... Its very helpful for me..

[@danielalejandro8045]

Gracefully made

[@kristakingmath]

Thanks, Daniel!

[@thatskap]

Fantastic video btw

[@ajaib1313]

Awesome.. Beautiful lecture!!

[@thatankahalawaye4582]

Thanks from France !

[@KF-cg6ey]

very nice video

[@umeshchaudhari7742]

plz solve my dout by single integration we get area then by double integration do we get volume than what by triple integration

[@pipertripp]

Good craic. Looking forward to the next one.

[@kristakingmath]

Thanks, Piper! :D

[@pipertripp]

Aye Cheers! While I've got your ear, integration in other coordinate systems would be a great topic. =)

[@rdaengineeringservices3543]

You just saved my 12 marks :D

[@PRADEEPYADAV-vs3rw]

Thanku mam, i never find such a nice explaination of double integral, once again thanku

[@kristakingmath]

Thanks, Pradeep! I'm so glad you liked it! :D

[@Jam7pi]

Nice work!!!! Thank you!!!!

[@kristakingmath]

Thank you so much! :)

[@mga5682]

Mam please tell me how to find volume of cone using double integral

[@h1a8]

Good video! What software/tool do you use to draw the diagrams for this video if you don't mind?

[@kristakingmath]

It's called Sketchbook. :)

[@ayodhyaprasadmishra8323]

According to yu double integration gives volume over the regions nd under the surface, but my teacher says tht it gives area only. it means double integration gives us the both value area nd volume as well. If yes then how? Please tell me it would be a great help

[@jesuslimon3885]

sheesh, amazing work

[@yasinsahin2962]

Soooo sooo thanks for this perfect video

[@Emc4421]

Hi Krista, you're awesome.

[@chamkaursharma6785]

Wow very nyc explained mam thanks