What does a double integral represent?
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- čas přidán 21. 08. 2017
- ► My Multiple Integrals course: www.kristakingmath.com/multip...
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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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. :)
Krista King good job 👏🏻 😘
Thanks!
thank you for this, my calc 3 class is taking off from here, so this is super useful
Krista King nice topi plz express triple intrigal
Awesome, I'm so glad! Good luck in Calc 3 this semester! :D
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....
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!
the best part about youtube vids like this is people gain value years after they are made. thank you !
I agree, Andrew, it's awesome! So glad this helped! :D
You've opened my eyes, literally. Thank you Krista :)
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
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😊
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!
Best explanation on yt by far... Great job Krista!
One of my NEW fav channels! Thank u!
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!
I'm so glad it helped!! :D
Great understanding+great concepts+great explaination+calm voice=great video.
Amazing work, i understood every single detail :)
Awesome!! :D
Thank you so much Krista, your videos are great and so helpful. You were born for this
Best presentation I've seen. Simple enough and very understandable. Keep up the great work!
😊
Not only did this explain Double integrals perfectly, it also summarized single variable integrals very nicely. Wish I had found your videos earlier!
Now this is what I was looking for. Feeling short of words to express my gratitude towards you. Excellent video!!!
Thank you, Krista, the explanation has a good stepping-difficulty. And colour-code is awesome too!
i cant believe how awesome this is, so clearly and simply explained. very nice!
This is the best explanation of an integral I have seen on CZcams
Thanks Krista! This was another great video.
Please come back Krista I need you to be my Calc 3 professor
Best overview in double integrals ever! And your voice is so soothing.
Thanks, Erika! So glad you liked the video. :)
You made it easy for me. thanks KRISTA .
Why this video is so underrated. Never see an explaination before. Thank you
I'm so glad you liked it! :)
This video gives you a good intuitive grasp of double integrals! Thamk you!
wow! I was not understanding the significance at all but ur pictoral explanations clear all my doubts.Thank u so much
Thank you very much for the wonderful presentation. Very insightful.
Your explanation is very simple and precise.Thank you for your lesson 😍😍
You're welcome, I'm glad you liked it! :D
Beautifully Explained...! Thank you for the detailed illustration of double integral.
You're welcome, Saurabh, I'm so glad you enjoyed it!
subscribed as soon as I heard your sweet voice, very well explained, you are an amazing teacher
Thank you so much, Blackfyre, I appreciate the sub! :D
Crystal clear amazing video. Very detailed.
Hi Krista, this is a beautifully sustained 25 minutes (and a bit) of clear exposition. Thank you!
Awesome.....now it makes sense why i should study more about them and i will.....thanks krista🖐️🖐️🖐️
Love your vids Krista; you and Rob Leonard are the best.
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!
You're welcome, Brandon, I'm so glad this helped!! :D
Can't thank you enough! You are an amazing teacher!!! Kudos to you! 😊
Thank you so much, Ketki! :D
Thank you Krista!
You're welcome, Sofia, thank you for watching! :D
I just say that I love you . Excellent ob .1st time some 1 clear the most difficult concept.
Thank you so much, Krista. I now have a different perspective towards multi-variable calculus.
You're welcome, Dev! I'm so glad you liked it! :)
I love your videos Krista!
You are great.
Your videos have been helping me since I was studying Calc I.
Wow, awesome! I'm so honored, thank you so much for your support! :D
I like it very much, It was very difficult to me to understand by the books, but your video just simplify everything, thank you!
You're welcome, I'm so glad it helped!! :D
Thank you Krista...I was wracking my head.. .it was a great relief!
You're welcome, I'm so glad it helped!! :D
you are a savior. Bow down to the math goddess 🙇♂🙇♂
Nice explaination.
I got cleared my concept,thru urr video.
Thanx a lot.
You're welcome, Shivam, I'm so glad the video cleared things up! :)
Awesome.. Beautiful lecture!!
Simply superb...Awesome...Please upload this kind of videos more ... It will be so much helpful to Mathematics lovers...Thank you so much madam...
If this lady had been my math teacher, I'd be a mathematician today.
Krista king and organic chemistry tutor are the best calculus teachers in the world ❤❤❤
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!
what a brilliant way to teach, please keep going
I will, thank you so much! :D
You are an amazing teacher, thank you!
Thank you so much, Carlos! :D
very satisfying video.thank you
Thanks, Vishal, so glad you liked it! :D
Awesome
I was stuck on this topic for like a week
Thank you 😊
You're welcome, Harsh! I'm happy to help! :)
thanks , now i really understood what is mean by double integration.
Thanks for the nice illustration!. Could you please share which tablet are you using here & which white board software?
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
Thank you very very very much, I understand this perfectly.
You're welcome, I'm so glad it helped! :)
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🤞
So glad it helped, Ahmad!
You have a beautiful voice, thanks for the video 😊
The way you explained was Extra-Ordinary !! ,Thanks for helping in understanding the double integral !!!.
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
You're very welcome, joliet, I'm so glad this helped clarify! :D
Outstanding!!
Brilliant work.
Thanks, Anup!
very well done.
+1 for sure.
Thank you so much! :)
Thanks from France !
I'm so grateful 🙏🏻🌷🌷
Very detail but concise at the same time. Would live to be your student.
Now, THIS, is how you introduce a subject; just wow.
*Thx Krista King for this vid; your explanation is pellucid and you are indeed erudite in mathematics.*
Thank you, Ivor! :D
Good craic. Looking forward to the next one.
Thanks, Piper! :D
Aye Cheers! While I've got your ear, integration in other coordinate systems would be a great topic. =)
Very good video !
Thanks you :)
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!!!
Nice work!!!! Thank you!!!!
Thank you so much! :)
Thanku mam, i never find such a nice explaination of double integral, once again thanku
Thanks, Pradeep! I'm so glad you liked it! :D
sheesh, amazing work
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.
You're welcome, I'm happy to help! :)
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.
Soooo sooo thanks for this perfect video
Gracefully made
Thanks, Daniel!
❤thanks a bunch
Good video! What software/tool do you use to draw the diagrams for this video if you don't mind?
It's called Sketchbook. :)
Thanks.. :) very helpful!
Fantastic video btw
Yeah.. Its really helpful.. Video is quite bit little lenthy but a very nice explanation has been given... Its very helpful for me..
I love your math lessons
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 CZcams.
I'm glad you liked it, I'll definitely be making more videos! :D
Wow very nyc explained mam thanks
You are indeed Krista King!
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!
I'll try to make one! And great point about the dA=dy dx... I appreciate the feedback! :D
that's fantastic - now what applications would that have in the real world other than finding the volume as you have demonstrated.
Hi Krista, you're awesome.
Wonderful! Thanks a lot... :)
You're welcome, Iliyan! I'm so glad you liked it! :)
Could we also see the volume under a multivariable function as the volume that the shadow of the 3D surface ?
plz solve my dout
by single integration we get area then by double integration do we get volume than what by triple integration
Thanks u for making such a great video!! :)))
You're welcome, I'm so glad you liked it!
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
Thanks you mum........ Love you... God bless you
extremely helpful
very nice video