Double integrals of type I and type II regions (KristaKingMath)

► My Multiple Integrals course: www.kristaking...
Learn how to use double integrals to find the area enclosed by type I and type II regions.
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If you could use some extra help with your math class, then check out Krista’s website // www.kristakingm...
● ● ● 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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[@McCloo57]

a 12 minute video is literally better than a 3 hour lecture. Thanks!

[@cfaion341]

your channel is a gold mine. donno how to thank you, seriously!

[@kristakingmath]

You just did! :D I'm just happy to help, so thank you for letting me know that you're enjoying the videos!

[@eamonbellful]

I pay thousands of dollars to take math courses at my university and yet I learn better and more from your videos. Not only does that show how great your videos are, it also shows a huge flaw in our education system.

[@watchmanling]

The sad thing is social mechanism is totally controlled by those big companies. If you want to survive, you have to find a job which means you have to have a degree, no matter how it is futile. Then your tuition for those education system support a huge amount of useless faculties who promote and brainwash generation by generation.

[@williamkoleszar3301]

This is why college maths education could be greatly improved if we just watched online lectures for everything and professors served as coaches/facilitators to guide conversations in class and help students with problems. In other words, students would watch lectures outside of class and inside of class would be one big group study. No more wasted hours staring a white board you don't understand.

[@leslieglen9047]

@@williamkoleszar3301 That's called the "flipped" classroom model, and many of us are using that in our classrooms. The downside is that many students DON'T watch the videos, and then our class time is spent trying to get them "caught up" and the students who DID watch and need my "coaching" help don't get it in the proportions that they should...

[@williamkoleszar3301]

@@leslieglen9047 That's true, there's one professor who teaches a flipped classroom at my university. Yes, we waste a lot of time catching people up. If this change will ever make a significant impact, students, parents, teachers, and the businessman making financial decisions would all have to be educated on the benefits of the "flipped" philosophy

[@mms1064]

Thank you My tuition money should go to youtubers like patricjmt and integralcalc. Not my professors!

[@nightmare4eVerr1]

I second that

[@financewithsom485]

True

[@levihammer9285]

True dat

[@hailey2822]

BLESS YOU!!! YOU JUST SAVED MY GRADE!!! seriously, no one else explained the differences between regions as well as you did, completely changed how I view this entire section!!!

[@kristakingmath]

I'm so glad it helped, Hailey!! :D

[@davidtiah]

I learned from you in 30 minutes what I couldnt understand from my lecturer in 5 hours. Thanks :)

[@dzarren]

Wait, the final answer of 1 is not actually the area of the regions. It is the volume of space below z=x^2 over the regions described. Just by looking, you can see that the area of the regions alone is actually 2, since we have 4 pieces that each take up half a unit of area. To obtain the area, you would integrate the function z=1, rather than z=x^2.

[@lucyguerrero6253]

Yess! Right, because there is a function inside the integrals. If it were just SSdA, then it would be the area, right?

[@El650Jefe]

I don't think she's calculating the volume either because z=x^2 does not form a surface. z=x^2 is a parabola which is two dimensional therefore it can 't have a volume. So i don't know what in the world was just calculated. If anyone know let me know.

[@omnimenge]

El650Jefe i think z=x^2 can be a surface in which the value of z is constant for all y..

[@omnimenge]

dzarren i agree, however, with z=1 we are actually calculating a volume of constant thickness of 1 which has the same value of the 2d area..

[@flyingsquirrel2620]

dzarren i agree. There isn't an area that equals 0,25. When we want to calculate the area of the triangle we should use 1 instead of x^2.

[@swethajeya574]

Thank you! my quiz is coming up soon! and this really helped me understand. College lecture notes are so confusing! thank u much! very clear!

[@kristakingmath]

You're welcome, I'm so glad it helped! Good luck on your quiz, I hope it goes great!

[@notSavant]

This is incredibly simplified which makes it easy to follow... thanks i love your videos, it's a shame there are some parts missing, but still, thanks!

[@FathimaMinna]

You helped me understand the portions that I spent 4 hours on without understanding a word. Thank you so much. I can't help but feel grateful! 😁

[@kristakingmath]

+Fathima Minna I'm so glad it helped!

[@ItsThatMilkshake]

Krista King has done it again. Thank you

[@moinaction]

I've always found your tutorial helpful and part of it is how you go in depth explaining every steps one by one. The way you teach, I bet a kindergarten kid could follow along and do well in the exam. Thanks for all your helps and God bless.

[@TheFarmanimalfriend]

I am finally getting a handle on double integrals. Your explanations are immensely helpful. Thank you.

[@Prodigy906]

Thank you for always being so clear and helpful! You explain everything so well and don't go too fast! ( : It really helps me as my Calc3 professor doesn't give us "steps" for how to approach problems only speeds through examples. anyway I really appreciate all your videos and the time it must have taken to do them all!!! ( :

[@kristakingmath]

+Prodigy906 I'm so glad I can help! Thanks for the comment. :D

[@hassanhashemi6478]

thanks, i was struggling with the concept.

[@kristakingmath]

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

[@oscarlykitty]

This is incredibly helpful, thank you! (It's also great to see women doing educational content on KZbin, so keep it up!)

[@webspierre9296]

Great job on helping calculus student with evaluating an interated integral and finding the area of a plane. How to use a double integral to find the volume of a solid region is another. Thank you integralCALC for the tutorial.

[@isaacmabitsela6864]

just came accros this while am busy with my exam prep. you are the best krista king

[@kristakingmath]

I'm so glad it helped, Isaac! I hope the exam went great!! :D

[@priMoN6]

You're the best. I wish you were my calc prof!!

[@lifdaai6736]

Thank you. Clarified the Type 1 and 2 definition very fast. Now it looks so simple

[@jakerupert3603]

This was so clear! Thank you for making these videos.

[@nathanfergot5710]

You did a great job on this explanation. Glad there are great teachers like you on KZbin!

[@whendidyoutubeaddhandles]

what my professors notes couldn't help me understand after two hours of frustration, you could within minutes. so helpful, I can't thank you enough :)

[@kristakingmath]

+chuck norris You're welcome, I'm so glad it helped!

[@nickvalencia147]

my teacher blazed over type 1 and 2 regions and i jusdt guessed what they meant all semester. no idea it was this simple, thanks

[@syuhada9627]

You explain it better than anyone! thankyou so much!

[@kristakingmath]

Aw, thanks Sara!! :D

[@Bigboinkies]

I was so lost in this portion, but you explained it like a King. Thank you.

[@rubyiskira8939]

Finally someone explained it properly, thank you!

[@mrmegahungry]

I LOVE YOU KRISTAAA! you saved me

[@kristakingmath]

So glad I could help!

[@Bluemonkey123]

You explain everything so well!! Way better than my prof!!

[@kristakingmath]

Thanks Karlos!

[@MikeRichards91]

ur videos have actually saved my life keep it up

[@kristakingmath]

+MikeRichards91 I'm glad they're helping!

[@yesman6097]

Your videos are all exceptional. You are a greater aid to my studies than my professor.

[@kristakingmath]

+Emilian Kachev Aw thanks! Glad you like them.

[@aliabdu6322]

Perfect , Thank you ... x^2 that's the main equation for the entire region( z=x^2) look at the first of the problem

[@raptormonkey253]

thanks for making this video, everything was explained really well.

[@guyyifrah849]

very clear video! Thank you!

[@kristakingmath]

+Guy Yifrah You're welcome!

[@Lonewolf-vd3yn]

Thank you! I never understood how to define superior and inferior limits when regions are II-type until I saw this video

[@harshitgupta2872]

to find the area of given region(4*D1) ,we just put the lower and upper bound for dy is from -x+1 to 1and for dx 0 to 1,it gives the area of region then why we put x^2 inside double integration ,isn't it like f(x,y) which give volume by applying double integration,I just want to ask what x^2 signified becoz it is not present /use anywhere in given figured

[@jacquelinezuniga3165]

You're so great at explaining concepts, thank you smart woman!

[@kristakingmath]

Jacqueline Zuniga You're welcome, I'm so glad it helped!!

[@Anna-qp1fi]

Excellent video, thanks for putting the time in to teach us :)

[@kristakingmath]

My pleasure, Anna! I'm happy to help! :)

[@ArchIVEDCinema]

These videos are really helping me get through college! Thank you so much!

[@aznangggieee]

Thank you so much for a very clear understanding! Way better than how my professor explains it :)

[@kristakingmath]

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

[@peiyingwu3036]

very helpful

[@lucyguerrero6253]

Hi! Your channel is super cool! Question, though. When you do the double integrals with x^2 inside, wouldn't you get a volume, though? Thing is, I'm using this for Probability and Statistics; what goes inside the double integral is the probability function, and when you do the double integral you don't find the area of the region but rather the probability, which is the volume of some solid defined directly above the defined region. Is this right?

[@brij1728]

thank you mam, you cleared my doubt.

[@TheCommander454]

Thank u so much that's really helpful .

[@kristakingmath]

Dhedan Saud I'm so glad it helped!

[@vychuck]

Krista Queen of Maths

[@HamzaM7891]

Amazing! Life saver!

[@BrandonOsborn404]

Concise and helpful, as always - Thank You!

[@kristakingmath]

+b dog You're welcome, glad it helped!

[@joliettraveler]

Great help in understanding this concept. I can't thank you enough.

[@kristakingmath]

+joliettraveler You're welcome, I'm so glad it helped!

[@rudzaninndwa6611]

thank you so so much. this is so helpful, I understood everything.

[@kristakingmath]

Oh good! I'm so glad it all made sense! :D

[@muhammadmohsin5194]

You're the star

[@indian_scouser_ynwa]

finally understood .thank u very much

[@kristakingmath]

You're welcome! Glad it could help. :)

[@akashi4470]

Uraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa yeeeeeeeeeeeeeeess Finally the most simple and clear explaining THANK YOU SISTER YOU ARE THE BEST

[@amalya94]

hi! i need help! I don't understand how you get the diamond figure D1, D2, D3, D4 from the given f(x). What if Im given a curve and a line how do I start?

[@EngGear]

thanks a lot, I'm grateful.

[@KoolViking3]

This video helped me so much - thanks!

[@kristakingmath]

+KoolViking3 You're welcome, I'm so glad it helped!

[@camf8646]

@dzarren To obtain the area of all four segments, I think you'd actually integrate z=1, and then take the absolute value of z=-1, and then add them together. Integrating only z=1 would give you the area of the segments D1-D4 between z=1 and the x-axis (1); integrating z=-1 from x=-1 to x=1 would give you the area of the segments D1-D4 between z=-1 and the x-axis. Similarly, I think you'd only take D1 and D3 into account finding the area below x^2, since x^2 never has values below the x-axis and therefore you couldn't find the area between the values of x^2 below the x-axis and the x-axis.

[@Mo_093]

Thanx so much.......That was really helpful !!

[@kristakingmath]

I'm so glad it helped!

[@SarahZeineldeen]

thanks alot! That sure cleared my misconceptions .

[@kristakingmath]

I'm so glad it helped! :)

[@MakroMat]

Krista hi I am math teacher in turkey and i am preparing math videos as you. Which program do you use for your videos? Can you say if you want please thanks

[@saqib3s]

Thanks Krista

[@Iwantblueberry]

I think you might have just saved my semester....

[@zackgrey4472]

You're my new calculus professor lol. Mantel being passed down from The Organic Chemistry Tutor, and that's how you knowwww you're good!!! Haha

[@kristakingmath]

I'm happy to be able to help!! :)

[@naveenvarma3359]

thank you for the video,it helped me a lot

[@kristakingmath]

+naveen varma Awesome! Glad I could help.

[@moshibudikgaogelo9383]

thank you so much. this was really helpful

[@kristakingmath]

You're welcome, Moshibudi! Glad it was helpful! :)

[@awsahmed6484]

you r a source of happiness!

[@kristakingmath]

+Aws Ahmed Thank you so much!

[@nawresalkhaffaf9257]

i know that the area of a triangle is half X base X height....the height and base equals 1 so we should get the are equal 1/2..... and the are of the 4 regions shoulb be equal to 2 not 1 ............am i wright or what ?

[@1495978707]

So, where does the x^2 come from? I know it's a given, but how do they find it?

[@chrismcdonald5118]

I believe it is your z equation, so z = x^2

[@younseoryu9542]

Yeah. You are for sure x100 better than my professor who has a phd degree from Stanford

[@CoCoW928]

Thank You..!!

[@kristakingmath]

You're welcome! :)

[@JeanDAVID]

I wonder why the area D1 = 1/4 ? is it not equal to 1/2 ?

[@shane2678]

You are the woman of my dreams. Beautiful & awesome at maths! ;) Anyway, creepiness aside, thank you for the video. It was really helpful. I understand double integration - particularly how limits of integration are worked out - much better after watching it. :D

[@strugglingcollegestudent]

Thank you so much!

[@harryrichard7106]

whoa that was amazing. You just saved my semester... thnx

[@kristakingmath]

+Harry Richard So glad I could help!

[@sudindras.8952]

krista thanks. the explanation is done very well. But area of a triangle is A=0.5*b*h, and in case of D1 its 1/2 not 1/4 can you justify please.

[@shehrozafzal9208]

you are amazing........!!!!!!!!!

[@EDUARDO12348]

Am a bit confused regarding the orientations of integrations, I thought we integrate vertically for dy not horizontally.

[@Littlejonjontheking]

So the difference between type I and Type II is whether you are looking at the graph from x or y.

[@gene546]

6:50, excuse me, ins't a right triangle what you're dealing with? I mean, you can use Pythagoras theorem to get the larger side right? Could you pleas explain

[@1972hattrick]

Intuitively I would've looked at the simple form before performing calculus. A 2x2 square with a sqrt(2)xsqrt(2) square inside leaving me with an area of 2. I'm guessing you should've taken the double integral of 3/2 x^2 rather than x^2.

[@TheHenkOwNzz]

How can a triangle (D1) that is half the area of a 1x1 square be 1/4 instead of 1/2?

[@VK-sp4gv]

I have the same question, like many others in the comments. I think the value of 1 she found is the volume under the surface z=x^2, not the total area of the four trangles, which 2.

[@johnvolpi125]

Hi Krista King. I tend to learn a lot from your videos, but part of this video confused me. I thought 1/4 would be the volume between z = 0 and z = x^2 in D1, but you said 1/4 is the area of D1? I thought the area of D1 would be 1/2. I got 1/2 by having 1 instead of x^2 inside the integrals. Am I missing something? Thank you.

[@haroldh6678]

Very nice, Thank you.

[@ethanszuba606]

💯saved my grade

[@ottoomen5076]

Could someone explain how the x^2 relates to the problem. I realize if you substitute x^2 with one you get the area of the triangle. What is the purpose of x^2? formulas like x^2 are given in my textbook but the connection between x^2 and the area of the triangle is never explained.

[@Siggfuggggg2000]

I don't understand why you put in the x^2 in the double integral after describing the triangular region

[@Money__Matters007]

very informative thanks a lot

[@leostrauss549]

Just curious, but you wouldn't happen to know about complex variables and calculus in the complex plane? Like contour integrals and residue theorem. I'm taking that course in the fall semester. Thanks anyways. Peace out!

[@Blacktop_Battle]

But shouldn't the area of all the triangles amount to 2? since the square is 2x2 (area of 4) and the triangles take up half of the square???

[@dweebler7179]

huge help thanks a lot for the help

[@kristakingmath]

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

[@victorserras]

Why use double integrals to calculate area in the first place? And you drew a 1 by 1 square and divided it in half, so shouldn't the area of the triangle be 1/2?

[@GH123XL]

Hi Krista, I wonder if you only concentrate on integral only and not cover topics such as tensor calculus. Thank you very much.

[@watchmanling]

Hi I can't identify the type for this area between a unit circle center at origin and another function y>x . Can you advise? Thank you so much.

[@chaimaedoudou1699]

Will you please do some exams you explanation is wonderful

[@joyanz114]

You are amazing thank u so much

[@maxwonkyi2546]

Thank you very much. I now understand😪😪

[@kristakingmath]

You're welcome, Max, I'm so glad it helped! 😊

[@sultnot2]

always perfect!

[@anthonykallingal4263]

Thank You, however when I am finding D1 using Type 2, I get answer as 5/12 instead of 1/4. Not sure where I am going wrong.

[@srpenguinbr]

Why can double integrals be either treated as area or volume of functions?

[@FaelFabriEngineering]

I finally got it

[@kristakingmath]

+Fael e kika Fabri Awesome!

[@javierrios4647]

thanks