Double Integration Example over General Regions --- two ways!

We introduced double integration originally for rectangular regions. Rectangular regions were nice because the limits of integration were all numbers. But what about if we want to calculate the volume under a surface over a more general non-rectangular region? In this video we look a full example of a general region and how we can think about breaking the region up into either little vertical strips or little horizontal strips. That is, we compute the double integration either first integrating with respect to x and then y, or switch the order. While this example worked both ways, in some double integral you will want to change the order of integration because only one of the two ways will be easy to actually compute.
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[@jamesjin1668]

I really feel that calculus is in fact based on very intuitive and practical goals, and through your tutorials, you've never lost that characteristic of calculus, thank you!

[@evelyntromp789]

I genuinely cannot believe I get to live at a time when this is available (and free!). Thank you so incredibly much

[@OMRIGREEN-xw3he]

I always come back to your channel when I want to get a more intuitive sense of a subject.

[@agh1750]

Absolutely phenomenal explanation, I'm going through your videos after finishing Professor Leonard's calc 3 playlist and I must say, although I love Professor Leonard for his "no man left behind" style of teaching where he almost over-explains everything, the slower pace results in a fragmentation of concepts whereas your concise approach fosters a more cohesive, elegant understanding of the material. I will say however, for people like me that aren't particularly good with spatial reasoning, going through Professor Leonard's playlist before attempting yours is the best way of avoiding confusion/frustration. As great as your lessons are, I don't think I'd be able to follow if it weren't for the preparation I did.

[@DrTrefor]

Thank you! It's true, this playlist is great for an overview and building intuition for the main concepts, but it isn't a replacement for an entire course.

[@MsLegobuilders]

I really just want to say thank you! I have my midterm tomorrow at 8 am and like this 2020 election got me all types of worried about my rights, so like thank you for eliminating some stress with your clear and concise explanations. No lie, you're better than my professor and teach really well especially for visual learners. Mad respect, I see you post so often about all the topics I'm learning about. I appreciate you and your videos a lot and just want to say thank you from the bottom of my heart.

[@DrTrefor]

Thank you so much, and I can't imagine the stress of having to write an exam when you are worried about the state of the world so much right now. Good luck!

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biden really fucked us over hm

[@CKPSchoolOfPhysics]

You are truly gifted sir !! Thanks a ton for teaching us mathematics in most visual way.

[@HermanToMath]

PERFECT TUTORIAL!!!

[@Sathrandur]

Having only previously studied calculus in two dimensions and wanting to learn how it could be applied to three, this video (which I found in the Calculus III playlist) has really opened my mind. It is really in essence beautifully elegant; despite how complicated it looks it is really quite simple. Thank you for enlightening me.

[@manhkhang7732]

Thank you so much Dr. Bazett for helping me through multivariable calculus. You are doing god's work and your love of mathematics is contagious. I cannot appreciate enough how wonderful and engaging your videos are. Once again, thank you!

[@nathanhunt4866]

Best channel at teaching the “big picture”. Thank you!

[@briandwi2504]

Lovely. Symbolab is an excellent tool with this subject. It will do the integration for you so that you can concentrate on understanding what is going on. Thanks for all your great explanations!

[@Rohan_Mahato]

Phenomenal explanation

[@sarvasvkakkar2545]

Sir the way you explain the concepts is awesome

[@DrTrefor]

Thank you!!

[@Cowboy_Consultants]

THANK YOU FOR THE INTUITION I HAVE SUBSCRIBED. This channel is super good

[@DrTrefor]

Thanks for subbing!

[@khushaalnandwani3372]

Thanks a lot, super helpful

[@TonyTorzido]

Thanks, homie.

[@luphiax4239]

Beautiful content and perfect explanation, keep it up

[@DrTrefor]

Thanks, will do!

[@hikmatullahpakhtoon3694]

I admire your way of explanation.

[@DrTrefor]

Glad to hear that:)

[@saurabhsingh-ow7ue]

thank you sir.........

[@anghaagarwal3970]

Sir please make videos of Surface integrals over scalar and vector fields

[@duckymomo7935]

I know Yu talk about Fubini’s here but did you mention it?

[@continnum_radhe-radhe]

🔥🔥🔥

[@muhammadumarsotvoldiev8768]

thank u

[@stephend.4342]

A significant teaching point that you missed is that for the example you worked through (integrated first with dydx, then with dxdy ) is that in that example you were not calculating the volume under the domain shown in the xy plane as projected into three space (in that case the intergrand would have been 1) but you calculated the volume with that same xy plane domain but as projected up into three space onto the infinite paraboloid 1 + x + y^2.

[@mousumimukherjee3516]

In the previous video you described double integration using Riemann sum. In this video there is a different approach. Is there a way to show that both the approach are equal?

[@holasanja6618]

what curve is A(x) the area under curve for?

[@Bokkie312]

Hey, your videos are really helpful and great but it would be nice if it was clear wat the x y and z axis are

[@meghnavarma5878]

what would we do if the region is bounded by non constant functions on all sides like a region bounded by: y=cos(x)-2; y=sin(x)+4; x=cos(y)-3; x=sin(y)+7 basically what would be the limits of integration when bounded by these 4 functions

[@NitinPandey-cv9wi]

❤

[@GastroenterologyPINNs]

👍👍

[@ricardobautista-garcia8492]

My understanding regarding A(x) is that as a specific x value is chosen, we will look at all the y values that multiply by f(x,y) (or z) to create that red area. The same is for A(y). Is this correct? Thank you for the video, it is great as always.

[@DrTrefor]

Indeed!

[@ricardobautista-garcia8492]

@@DrTrefor Thank you.

[@divyeshvankar4818]

why limit in 2nd method is not from 0 to y^2 ?

[@cyndimouse827]

Why are the bounds for the horizontal approach still the same? The y=0 lower bound makes sense to me because that it's given. Why wouldn't the upper bound be sqrt(x) instead of 1?

[@madghostek3026]

Well, to get all the horizontal strips possible, you must go from x=0 all the way to where its allowed (under the sqrt(x)), it intersects with x=1 which is at sqrt(1)=1, looking at intersections points on the shadow helps a lot.

[@adnanhussain9906]

2:25 can anyone explain what he means by "I imagine integrating all these values of x" how does integrating A(x) with respect to x give us the "sum" of each of our individual A(x)s for fixed values of x

[@madghostek3026]

Remember that questions about area and arclengths etc rely heavily on riemann interpretation of integral, here you have A(x) which is area of a plane, we don't know yet how this works, but once magically you get that area, you can multiply it by change of x and you get volume of some cuboid. If you take small enough steps (the dx) and sum all the cuboids (integral of dx)x you can actually get the whole volume

[@vijaysinghchauhan7079]

Please do not use the red color it gets hard to read.

[@usmanhaider295]

Sir plz use less pronunciation....some students can't get you clearly

The English is perfectly understandable to those fluent in the language.