Riemann sums to approximate volume of a double integral (KristaKingMath)

► My Multiple Integrals course: www.kristaking...
Learn how to use Riemann sums to approximate the volume that lies above the given rectangular region defined by the given x-interval by the given y-interval, and below the given function. You'll divide the interval into rectangles and then find the coordinate points that define the upper right-hand corner of each of those rectangles. Then you'll evaluate the original function at each coordinate point, take the sum of those values, and multiply that sum by delta A, which is the area of one of the rectangles.
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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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[@MrJoeHelmet]

6 years down the line, Corona times, it is still helping students. Thank you!

[@kristakingmath]

You're welcome, MrJoe! :)

[@mertozarslan8925]

As an engineering student, I would like to thank you that these are so simple and detailed just like I wish to study for :D

[@kristakingmath]

So glad you like them! :)

[@hg2.]

This is another I to sort of take "on faith" and that all will be revealed by Krista in later videos. Krista is basically the only math teacher you can put that kind of trust in - the rest of the industry- forget it. Looking ahead there's lots of titles dealing with Double Integrals, which makes sense about where the action is and where this is headed.

[@renoir8]

You are saving my life this summer. Thanks so much.

[@kristakingmath]

le bandit du kwat You're welcome, I'm so glad the videos are helping!!

[@bradmotors3826]

as an engineering student, I appreciate all of these calc 3 video tutorials! You have saved me countless times on my tests!

[@ekeneduruaku5103]

I love u! This helped so much! I find it great how ur work from years ago still helps students now.

[@kristakingmath]

I'm so glad it helped! :D

[@Animeshuma]

​@@kristakingmath Hi Ma'am how do you make such videos? On which software I mean to ask. Can you please help?

[@gammadas]

I really enjoyed your presentation--I watched this with my daughter and told her to lay out everything step wise as you did to get partial credit in case of a silly mistake. I took a lesson in teaching/presentation skills as well. I look forward to more

[@kristakingmath]

I'm glad it could help!

[@MuccisJag]

Thank you for this! I missed the lecture because I was depressed. This really helped!

[@davidization6]

Lifesaver!!! I can't believe how easy you made this, my textbook sucks!

[@kristakingmath]

David Park LOL, awesome! Glad it helped!

[@jonathangranda3490]

Good review before tomorrow's test! Thank you!!

[@kristakingmath]

Glad it was helpful, Jonathan! I hope the test went great! :D

[@Eng.HassanElwy]

Thank you so much, my amazing teacher ❤️

[@kristakingmath]

My pleasure, Hassan! :D

[@salami19-v4t]

very clear explanation, thanks

[@mrnarason]

Thanks! professor didn't each teach this but assigns these problems.

[@kristakingmath]

+Victor P. I'm so glad it helped!

[@PK-wr6kz]

Clearly and thoroughly revealed. Been a mystery up to now. Thanks!

[@kristakingmath]

You're welcome, Peter! I'm so glad it made sense! :D

[@douchelip]

Excellent detailed explanations. Your videos are invaluable. I hope you continue making them.

[@kristakingmath]

Thanks! I plan on it. :)

[@usmankhokhar8083]

I,m going to many websites or watching many tutorials but this toturial so helpful save alot of time thank you so so much

[@kristakingmath]

+usman khokhar You're welcome, I'm so glad I can help!

[@alexc5448]

truly thank you from the deep of my heart

[@kristakingmath]

You're welcome, Alex! I'm so glad the video helped!! :D

[@learntoday7674]

This is Excellent ❤️.. Well Explained... Thank you very much.....💯

[@kristakingmath]

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

[@RSLT]

Great Video. Very interesting and informative. Great Job!

[@danielm3569]

I am taking this class online right now. Thank you for being my teacher.

[@kristakingmath]

I'm glad I can help along the way!

[@Animeshuma]

Lovely explanation.. crystal clear everything!

[@kristakingmath]

Thank you so much, Animesh, I'm so glad you liked it! :D

[@rajatsaini9275]

thanks helps so much

[@luvyadav3853]

Love u crista, its helped me.

[@kristakingmath]

I'm so glad it helped! :)

[@paddy.flavin2805]

what if you didn't subdivide it into four rectangles and only did it with the one whole rectangle would the values for m, n, i and j change?

[@maudentable]

Highly informative.

[@gu6252]

Brilliant explanation😆😆

[@FreedVoid]

Explained it well :) Thank you very much

[@jeffreysucuzhanay4695]

very clear! thank you so much!! coming in clutch haha

[@kristakingmath]

So glad it helped! :D

[@mckamike]

Wow just what I needed, thanks

[@kristakingmath]

+Legolas Longleaf Awesome!

[@ashanpdw]

This really helped. Thanks! :D

[@deedakk1563]

thank you so much

[@real01s.50]

what if m = 3? would you divide the rectangle into three lines? I don't understand. your response will be helpful, thank you.

[@Yourlocaljp]

Yes, the rectangle would be divided into 6 subrectangles, it would be a 3 by 2 or 3M by 2N where M is x axis and N is y axis. Three rectangles touch the X axis and two rectangles touch the Y axis. 3M by 2N looks like clenched teeth NOT like a six pack of abs. 3 length 2 high ^ Then you would use 6 points to approximate the volume by evaluating f(x,y) at each upper right hand corner point. The delta A or change in area would be the change in X times the change in Y, or just the area of the bottom-most left-most rectangle that is touching zero (0,0).

[@blachardia]

Very helpful, thank you so much!

[@kristakingmath]

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

[@jr52990]

If I just use the R equations values into the double integral would I get the same answer?

[@dennisbor-iq9lp]

Very clear

[@kristakingmath]

Glad it made sense!! 💪

[@MikeRichards91]

very helpful thanks for uploading this

[@kristakingmath]

+MikeRichards91 You're welcome!

[@VikIsLost]

thank you!

[@kristakingmath]

+Kip Dynamite You're welcome!

[@josephanthonycohenbrownsto8378]

What about in spherical?

[@lesegomokgabudi5798]

THANKS

[@kristakingmath]

You're welcome, Lesego!

[@brandoncrocker4763]

So if m was 3 instead of 2, would you then divide the x distance (which is 2) by 3?

[@powerjustice1444]

so helpful.. thank you.

[@kristakingmath]

Glad it could help!

[@zaynabs5711]

Thank you so much for your videos, so helpful!!

[@kristakingmath]

+Zaynab S You're welcome, I'm so glad they're helping!

[@yufenghe8285]

why the five non-green points, i.e. (0,0), (1,0), (2,0), (0,1/2), (0,1) were missed from your summation? Since you added one extra point for both x and y, which should give you 3*3=9 terms for the Riemann sum.

[@xxdriftking027xx]

+Yufeng He correct me if I am mistaken, but in this problem, we are calculating the volume above the rectangle using the [Upper Right Hand Corners], which was defined at the beginning of the problem. So in the summation, we use the (x,y) points of all the upper right hand corners of the four rectangles. Since there are only 4 rectangles, then we should only being using 4 points in the summation.

[@yufenghe8285]

+xxdriftking027xx Right. Thanks, I think you are correct. However, another simple question that pop up in my mind is that, why the points from upper-right corners? Apparently you have four options which may give you different results. In this case, the answer would be very different if you chose bottom-left points. There must be reasons behind the choose of points for summation. Please let me know if you have clues.

[@completelystupid87]

awesome, save me 30 min reading the book, i wish you were my prof, mine just sucks

[@kristakingmath]

+completelystupid87 I'm just glad the video helped!

[@hg2.]

Yes, Krista is emptying out those miserable math faculties world wide. But don't expect to see any tuition reduction from the education racket.

[@hg2.]

+hettygreene ... that's not fair- math teachers aren't all bad. But Krista IS making all of them obsolete - this is simply better than the classroom .

[@GametimeBrizzle]

So, if I was asked to find the volume that lies BELOW the surface and ABOVE the rectangle (with same m=n=2), could I just use the lower left corners for my sample points and solve the same way (just with the different sample points plugged in)?

[@miamjs3398]

Thank you so much for your videos

[@kristakingmath]

You're so welcome! Glad I can help!

[@user-dd8we9kv2l]

I still can't get past the notation completely...

[@chahilramharakh7165]

thank you!!!!!!!! you are amazing :D

[@kristakingmath]

+Chahil Ramharakh Thank you so much! :)

[@GametimeBrizzle]

That doesn't seem clear after I read it... Basically, does the relative position of one's sample points (LoL,LoRt,UpL,UpRt)dictate whether the answer represents the volume [above||below] the surface and [above||below] the rectangle? Or does the selection of points not matter as long as the choice is consistant?

[@kristakingmath]

when you're using riemann sums, you're almost always approximating area ABOVE the rectangle and BELOW the surface. depending on which sample points you use, and the way the surface is oriented, you might get an overestimation or an underestimation. i hope that helps! :D

[@fouried96]

+Krista King | CalculusExpert.com But isn't z = 0? So shouldn't the volume sitting above the rectangle = 0 as the 'height' (z) = 0? Thanks

[@motsamaimokoena208]

Wow.....just wow

[@l3yonhavip]

why professors always like to make our life hard :) . thanks your video make it easy

[@kristakingmath]

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

[@heatherlooney1376]

is this for collage i am in middle school

[@heatherlooney1376]

i am 12

[@heatherlooney1376]

i need help with Dimensional Analysis

[@kristakingmath]

This is for Calculus 1 or 2. Some people cover it in high school calculus, and some get to it in college. :)

[@notsurewhattoputhereyet7052]

so i just realized you have discolored eyes, that's cool

[@ncape6486]

these videos dont teach anything. they just take one specific problem for one speicfic case and just solve th eproblem. not helpful at all.