Defining Double Integration with Riemann Sums | Volume under a Surface

We generalize the ideas of integration from single-variable calculus to define double integrals. The big idea in single variable calculus was to chop up the region into a sum of little rectangles called the Riemann sum which was an approximation for the area under a function. Then we took a limit of the Riemann sum to define the definite integral. We do much the same here, looking to find a formula for the volume under a surface. Now a rectangular region in the domain is broken up into a lot of little prisms and the sum of those volumes is the Riemann sum. Take the limit as the sizes in that partition goes to zero and this defines a double integral.
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[@DrTrefor]

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DUDE. this was the explaination i was looking for. a lot of other profs just repeat the formula, i want to know HOW and WHY it works. Thank you so much!

[@DrTrefor]

Glad you liked it!

[@EldenEngineer]

This is exactly right. My professor doesn't even draw any of the shapes or explain what double integration actually does for us. Why of why is American university education this way?

Haven't seen a better explaination of double integration than this. Well done!

[@DrTrefor]

Thank you!

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I'm in the middle of the video and I can't wait to comment, I'm so grateful for your helping me understand the crazy concept in such a beautifully simple way, thank you!

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:D

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Thank you again for the great explanations. You are doing great work and it is very much appreciated. I really love the visuals and how everything gets linked with the graphics. It makes it much more clear for me. I even would like to see them linked in even more when you are doing functions and such. It makes the concept so much clearer. Thank you. Wonderful.

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Thanks again, Dr. Bazett. Whether I'm preparing for an upcoming class, or sifting through yesterday's lecture, you always help me to understand the concepts and appreciate the **elegance** of calculus. Now for the heavy lifting...

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Your voice reflect that immense passion for mathematics

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Being someone who hated math throughout high school, studying it at uni was a huge block for me. Your channel has helped me a lot! Falling in love with calculus ❤ thank you very much

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Taking Calculus 3 and this was extremely helpful! Thanks!

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[@lubnaabbas7329]

amazingly vivid explanation! love this, this is really missing from our lectures at the uni..

[@yaboi1525]

Thanks a lot. One of the best interpretation of double integrals I've encountered. You are a great teacher.

[@DrTrefor]

You're very welcome!

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Hats off to you for explaining like this.

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[@notsnowman]

This was a great video. thx!

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Thanks bro,you gave me clarity what exactly double integration mean

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Thank u sir...... it helped me a lot ......

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Very cool sir.

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[@DrTrefor]

Thank you!

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[@DrTrefor]

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thank you so much you are so good at explaining!!

[@DrTrefor]

Glad it was helpful!

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Your videos are awesome. I didn't find your channel until I reached calc 2 because for some reason your videos don't show up when I search for calc courses

[@adamsleep3028]

I was actually looking up videos of big math channels on double integration, but they all focused more on computations. Yours is way more explanatory of how double integrals work, you deserve way more views than this.

[@maximusthiers698]

I watched 2 other peoples videos on the topic prior to this one and I can only describe their attempts at teaching this as "absolutely useless" in comparison. Good work my man.

[@DrTrefor]

Thank you, glad it helped!

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Wow what a wonderful video. Thankyou so much ❤️❤️ So underated. Your videos are the only helpful ones i found

[@DrTrefor]

Thank you so much!

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@@DrTrefor Thankyou so much too. All the other videos are either too formal and lengthy and show no intution. And on other extreme. Khan academy only showed intuition but didnt help me with my college course. Your videos cover everything thoroughly and are very enjoyable with great energy. ❤️❤️❤️

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First comment, every video you make is just perfect :)

[@mono7891]

Great video ! love the way you explain. Most of the professors in the math department cannot explain the physics let alone showing the 2D or 3D representations.......Good that we have tools like MatLab/wolfram alpha or any CAD/Finite Element code ...

[@DrTrefor]

Glad you enjoyed!

[@user-rb7vn3lt8t]

Thanks a lot ,prof :)

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:)

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[@DrTrefor]

Glad you liked it!

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thank you sir

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Thanks sir

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[@DrTrefor]

You're welcome 😊

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[@SuperDeadparrot]

I’m wondering if rectangles are the best shape to use for this.

[@HuyNguyen-fp7oz]

finished!

[@aidenstonehouse9651]

Would it be possible to see how the limit definition works with an example. Similar to how you did for the single variable case where we find an expression for x_i and deltax in terms of n, and then let n sum to infinty?

[@sinasoltan.m4859]

how similar and how different are vector functions and vector fields? this question is blowing my mind:/ in both cases the domain is a subset of real numbers and the output is a vector but why do we draw the output vector of a vector valued function from the origin (0,0,0) and the output vector of a vector field from its domain ( a particular point like (x, y, z) ) I know it might be unrelated to this video but I will be thankful if you answer my question😊💜

[@Julian_MacKinnon]

Your 3D Riemann-sum plot is amazing, were you able to generate it in TikZ? I've been struggling to make a good looking diagram for the concept

[@JaydenTunde]

Would that mean the "n" that is used to compute ∆x and ∆y are the same?

[@commandermakki255]

0:04 Did you mean the volume under a surface here or the surface area of the shape? And if we're finding the volume under the surface, isn't this similar to single variable calculus where we used to rotate the shapes over an axis and find the volume of the solid created, is the only difference that the shapes created in single variable calculus are symmetric while shapes over here are not necessarily symmetric? I just want to make sure I got the idea.

[@anhimanyu1]

Hello Trefor, thank you for the explanation, just wanted to ask why is the change of x and change of y =2 and not 1??

[@fernandotanase114]

Pretend that "m" stands for the # of partitions(cuts/squares) along the x-direction, and that "n" also stands for the # of partitions but in the y-direction(in this case m=2,n=2). Also, let's say that the Domain of x is [a,b] & y is [c,d] (in this case x:[-2,2] y:[-2,2]) . You'll find that change in X= (b-a)/m & change in Y=(d-c)/n. In this case change in X =(2-(-2))/2= 2 & change in Y also equals 2.

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❤

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[@tanish6035]

But sir why we took only four squares in last example??

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[@CI-ym5hr]

2:38

[@shiveshsingh193]

how to find f(xk,yk)

[@Mlridge]

Doesn't it give the volume of just one rectangular object under the curve? Shouldn't we multiply it by 4?

[@andrewkoulogeorge2569]

what is your background in mathematics?

[@DrTrefor]

Algebraic Topologist:)

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Chalk on the shirt is sexy. And inspirational video, as usual

[@donotbebiased6987]

sir fundamental theorem of line integral, green theorem, etc r part of multivariable calculus course so y didn't u add these videos to this playlist

[@DrTrefor]

those are in my vector calculus playlist:)

[@donotbebiased6987]

@@DrTrefor thanks professor

[@thesoul3461]

Has anyone noticed that the little high pitches in his voice sounds much similar to Grant Sanderson's?

[@qiping7165]

5:36 I think the sigma notation should be a double sigma to cover every point on 2d dimension.

[@mathrovert]

I've seen it both with single and double summation notation. I think with single it's meant to apply more generally to a shape with an area.

[@user-mg1hz2qm8k]

HALLELUJA 💖💖💖

[@jonpritzker9709]

0:03 ... the *volume under a surface?

[@zurikodzuliashvili9556]

Thanks, it helped me a lot. may i know what software you used to create graphics of function with rectangles in it ?

[@waldenfreedman3457]

At the very end you said that the value 80 was an approximation to the "area" under the surface, but you surely meant volume. Not to be nitpicky, it's a good video otherwise.

[@DrTrefor]

Great catch, thank you!

[@VK-sp4gv]

I had exactly the same question. Am I right then to say that if f(x, y) is a constant function 1, then the double integral will give the area of the region? Also, is this region the projection of the surface f on the xy plane?

[@waldenfreedman3457]

@@VK-sp4gv In terms of the numerical value, the double integral of the constant function f(x, y) = 1 gives the area of the "base", but in terms of the units, if all distances are measured in say meters, with the height 1 meter, then it would be numerically the area but its units would be units of volume. For example, if D is the unit disk and f(x, y) = 1 (meter) then the double integral of f over D would be pi cubic meters. And certainly the projection of the surface f(x, y) = 1 onto the xy-plane is the 2-dimensional region D (the base of the solid.)

[@michaelempeigne3519]

why not just say n -- > infinity

[@ANTOINETTE-nk1tm]

CALCULUS IS THEE GREATEST MATH EVER INVENTED OR DISCOVERED BY MANKIND.