Calculus 3 Lecture 14.3: Double Integrals over POLAR REGIONS
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- เผยแพร่เมื่อ 26 ธ.ค. 2024
- Calculus 3 Lecture 14.3: Double Integrals over POLAR REGIONS: How to perform Double Integrals over regions using Polar Coordinates/Equations. How to translate Rectangular Double integrals into Polar Double integrals.
Ignore this. It's for myself but use it if you find it useful. :D
Intro to double integrals over polar regions~ 0:00
Volume over polar region examples~ 22:45
Ex. 1~ 23:55
Ex. 2~ 42:15
Ex. 3~ 53:50
Ex. 4~ 1:03:00
Ex. 5~ 1:23:41
Side note about theta not always being from 0 to 2pi~ 1:34:51
Example of polar region where theta is not from 0 to 2pi~ 1:38:00
Explanation of how to determine theta bounds carefully~ 1:46:10
Volume between two surfaces~ 1:59:04
How area can be represented by volume~ 2:37:17
Ex. 1- (Area of region bounded by one polar function)~ 2:42:17 (choosing theta bounds~ 2:47:16)
Ex. 2- (Area of region bounded by two polar functions)~ 2:53:28
More volumes in polar form~ 3:01:03 (choosing theta bounds~ 3:07:59)
Volume using polar coordinates on a rectangular region~ 3:16:19 (used when function (integrand) inside is difficult)
Thank you so much I did not want to spend hours skimming for one part
On
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These lessons may be long, but they are such high quality. People pay money to get these lessons elsewhere. Thank you so much for making these lessons available for everyone! you saved my grade in cal 3.
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shin akuma I got annoyed by that ad as well.
These lessons are long because they are incorrect. The extra r makes it physically impossible for a double integral to return an area^2. I wish professor Leonard would not continue to try to justify this misinformation. I am used to seeing stuff that is wrong and figuring it out from getting my degree in Biochemistry.
These lessons are long because they are incorrect. The extra r makes it physically impossible for a double integral to return an area^2. I wish professor Leonard would not continue to try to justify this misinformation. I am used to seeing stuff that is wrong and figuring it out from doing research for my degree in Biochemistry.
@@TheFarmanimalfriend what? look up the actual proof of polar integration and you'll still see the extra r.
"some of you guys arent even watching" these kids taking the legend for granted :(
maybe some students love to study in relax mode rather than in class
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Professor Leonard. omg, today I had my exam on this chapter. I went through all your lectures. Did your problems over and over. And today that i had my exam, I feel I was able to do all the problems. I feel confident. I dont know my grade yet. But thank you so so so much.I feel like i learned with you. Youre the best!
update on your grade !!!
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Honestly this professor teaches multi variable so much better than my professor at duke. But i guess its because we are trying to cover the same thing by less than half of the time
Still, this lecture is so helpful to me.
same problem here!! great professor, but we only get 3 hours a week...
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1:46:15 Anyone else agree that the Theta Limits are -pi/2 to pi/2 instead of 0 to pi. It does work out to same answer of 9*pi
That's what I wrote prior to watching his explanation. I haven't confirmed that it's equivalent yet.
yea that also works too
@@yinkak3921 Isn't that what it _should_ be, though?
It seems very weird to integrate over a quadrant where the region doesn't even exist - that goes completely against pretty much all other double integrals that I have ever seen.
@@Peter_1986 in polar coordinates when r is negative, the point get's flipped so for example if theta = pi and r = 2cos(theta) then r = -2. since r is negative we flip it so the point actually ends up (2,0) which is on the region. He talked about how to plot points in polar coordinates in his calc 2 playlist so you might want to watch that
yeah it works and later in the video in a similar example he used -pi/2 to pi/2 instead of 0 to pi
Just want to say Professor Leonard, your calc lectures got me through some really rough spots in my undergrad math classes. You’re a gift, and your lectures are clear, to the point, and made what made no sense in class somehow just click after watching you teach the same topics. God Bless you!
x=r*cos(theta) and y=r*sin(theta). r first, then theta. dA is gonna be r*dr*d(theta)
circle involved? use polar.
43:00
50:40
***1:05:00 (sum of squares)
***1:22:10
be careful with theta. 0-2pi doesn't always give you the area of the circle exactly once
***1:39:00
1:59:00 => to find volume between two curves f1&f2, you gotta find area trapped between them. this can be determined if you know where they intersect. f1(one on top) - f2(one below) [sub a point in the relevant region and find out which one has greater value]
2:15:40
I hope these videos last forever, I cannot express the amount of gratitude i have for you making these videos and the impact they have. You have answered all the questions I had that my teacher was unable to convey clearly to me. Thank you so much for these videos!
13:38 "Take your ZONE-IN pills... I don't even know what those are" He tried playing off like he doesn't know what aderall is
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watching the video on ritalin :D
I love how he says that he wants us to solve for the integral on our own and then he picks up his marker and solves the problem for us. Leonard, you are amazing! THANK YOU!
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Thanks heaps for professor Leonard's lessons, I really appreicate it and they help a lot! Though at 2:47:27, when theta goes from 0 to pi, the result of the double integral is 9pi/4.
Its a long section but every minute is worth the watch, lots of good info. Whats really sad is the last 2 videos are roughly 7 hours. My professor spent one class of 50 min to go over both sections. Then he sends everyone an email saying he is disappointed with everyone when the average test grade is 50-60% and that we need to do more homework.
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The setup at 1:46:02 is technically correct, but the integral makes more sense if you integrate theta from -(pi/2) to (pi/2).... they give the same result, and I feel this is far more intuitive geometrically.
I was really struggling on this topic. And I searched about this before my quiz, hoping to get some sample questions related to this topic. But lucky me got whole lecture that too in free of cost. I paid thousands of dollar in my university and did not understand anything but this video made this topic really easy for me. Kuddos to Professor Leonard. And I would like to thank him from bottom of my heart. Thanks for this video!!
Anyone else listening at 1.5-2 speed to try to get through more lectures faster
Yes. The whole thing is easy. I just slow down when he wants to prove something.
that's the only way to watch TH-cam Math videos.
Is that beneficial
I did but I noticed that when you really take the time and understand what he says instead of just doing the exercises, you will learn in a much meaningful way. I advise you to stop doing that guys
@@kozukioden2406 theirs objective is to crack exam, not to become lecturer, but as a student one must give full attention atleast to what he says, pondering on his points is far away.
Professor Leonard thank you for another monster and lengthy video/lecture on Double Integrals over Polar Regions. This is a mammoth amount of material for any Math/Engineering students to absorb , however deep pattern recognition and practice wili help in all levels. From reviewing this topic and taking notes, I will rewatch this video for a clear understanding of the material.
Thank you for this Professor Leonard! With the recent jump to digital classes, my math professors already difficult lecture style has become even more difficult but your videos are helping me get through it. Thank you for saving my grade this semester ^^
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23:00 Start ex 1
42:10 Ex 2 Between z=x^2-y^2 and cylinder
54:30 Ex 3 (Good one)
1:03:00 Ex 4
1:24:00 Ex 5
2:48:36 Why would you go from 0 to pi anyway? The angle goes from -pi/2 to pi/2 and using those bounds gives the correct answer.
Omg thank you i didnt understand why either...
Can i know why thougj im confused
@@ozurking4748 So, it happened that the surface was symmetrical along that axis, but it wasn't determined by the area of the region. I hope that is clear enough.
Wouldn't integrating with the bounds -π/2 to π/2 give us zero? since π/2 to 0 and 0 to -π/2 are on opposite sides of the x-axis? Correct me if I'm wrong (four years later)
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For those who need help with the integral he calls "nasty" at 1:57:00 - it is not nasty. Use double angle to evaluate cos^2 and cos^4 (for cos^4, use the double angle formula and square both sides). The algebra ends up being nice - not sure why this guy made a thing out of it lol. Standard Calculus 2 integral
I'm actually doing that arrrrrrdrrrrrdtheta every time now out loud.... Thank you so much for having these videos up! You are the reason I understand calculus despite sitting through the actual university courses. I would be so lost without your videos.
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In the problem at 2:45:00 you take theta as 0 to pi/2 and double it. However, there is a similar problem at 1:45:00 where you take theta as 0 to pi. Why can't we take theta as 0 to pi/2 and double it in the problem at 1:45:00?
I want to thank you so much! so lost in class but i came across your video and I understand! This got me so motivated right now, God bless you.
At 1:48:51 when he chooses theta to go from 0 to pi, could you also choose theta to go from -pi/2 to pi/2?
I had the same question. Yes! After painstakingly solving the integral, I calculated the final answer using limits 0 to pi and limits -pi/2 to pi/2. They both led to the same answer. However, my answer was 5pi instead of 9pi. I've checked my work a total of three times, and I've checked my integrals using an online integral calculator, so I'm positive my answer is right. Nonetheless, it doesn't matter if the answer is 9pi or 5pi because your question has been answered.
Thanks! I'm glad it worked out because it feels a lot more intuitive to choose between -pi/2 to pi/2 than 0 to pi :)
Right? Even at 2:48:15 I feel like you can use -pi/2 to pi/2 as your limits instead of 0 to pi/2 and doubling it. But I'll let you check this one for me lol
I have to agree. You are not the only one :)
i had the same idea . but i wanna be sure . is that a real thing or a coincidence ?
شكرا
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These lecture videos are pure gold, thanks alot Professor Leonard
At 34:30, why would it not change to -π/2 to π/2?
35:21 for anyone who didnt understand where did 1/1 came from..........for x axis 0 + 2 / 2 which is 1 for Rkx.........0 + 2 / 2 which is 1 for Rky......tan theta = (Rkx / Rky) tanm inverse of that value will give you the starting angle
I'm 99% sure Professor Leonard made a little mistake explaining that you cannot go from 0 to pi on the area integral at 2:48:40 . He specifically said it himself on 1:57:00 that doing that is okay because the negative will cause the integration to "flip" onto the side that is part of the region. Also, I tried the integration that goes from 0 to pi myself and on the integration calculator and still got the same answer; 9pi/4.
Yeah I noticed that too. But maybe he just wanted to make a point about exploiting symmetry.
1:06:45 should ot we assume that we are in the first octant to choose the upper part of area not the one in the other side ? I AM REALLY CONFUSED
These videos at 1.5x are so entertaining and still understandable as review. so glad i have mtn dew.
i watch at 2x
2x at excercised and 1.5x at concepts
Minute 2:49:00
Here and in the previous example, i would take the angle between -pi/2 and pi/2...the minus will turn the area below the x-axis from negative to positive.
The integral at 1:49:00 is very unintuitive.
Practically everyone would feel much more comfortable with taking the integral from -π/2 to π/2.
Thank you very much Prof; you really know how to make my academic life easier.Next week I will be sitting for my examination on this concept, I'm now ready all because of you.
Hello professor Leonard can you do "differential equations". I like how you explain the problems and you covered everything which helps a lot. I just like to know more math. Just what I learn in class is not enough for me. And thank you for all of ur videos
Thanks Prof Leonard, I am seriously ahead because of you.
Watch me crush the final exam on 5/27/2022.
Interesting. I'm doing Calculus 3 and i'm 11. I'm also a contortionist and I find Calculus super easy. I've moved on to Stokes Theorem and Navier Stokes equation and Schrodingers equation and Gauss Theorem and Newtons theorem and Greens theorem. I love math. I'm also doing Triginometry and Calculus.
I fell asleep watching something completely unrelated and woke to this. Not mad. 10/10 will save this for later.
Why on 1:49:18 the theta bounds are 0 and pi but in 2:48:20 the theta bounds are double of 0 and pi/2?
Can't we take theta from 0 to pi in the question (3:08:31) just like we did in question (1:46:25) or can't we take theata from - pi/2 to +pi/2 in the previous question (1:46:25)?? 🤔😕
Thank you so much Professor
Teachers like you made me fall in love with maths
Keep up the good work
r^2=2r cos {θ}, r=2 cos (θ), i got it now thanks Leonard 👍👍👍
Thanks! One question on : 1:50:02 why does tetha go from zero to pi. Why doesn't it go from minus pi/2 to pi/2. Thank you!!
One of the most sensible videos on double integration
prof you have made my life easier.thanks oncemore
00:42 If the regions are circular use polar. If they are lines and rectangles use rectangle coordinates. What are rectangular coordinates? What are polar coordinates? You have an angle, go across to another angle, between 2 functions.
3:00 If angle between 2 constant angles then.. Q: Why is theta always last? A: theta = to constants everytime, thus d(theta) goes on outside. Why is theta always last?
Reference: x=rcos, y=rsin, ... watch 11.6
5:25 - 2 Cases. Case 1 - volume over a polar rectangle. Case 2 - volume over a general polar region.
8:15 - *FUNCTIONS FIRST* CONSTANTS LAST. In case 2 instead of hitting a (a rectangular region) you hit a function instead. How do you write a function in our limits? In this case its r=a function with respect to theta (r=g(theta))
east or west professor leanord is the best teacher
You are AWESOME SAUCE!!! You make math GREAT AGAIN! Thank you for not being boring!
13:30
they're called Adderal.
Please do differential equations Professor Leonard!
BEST PROFESSOR EVER. Thank you for all you do!
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thankyou soo soo much sir , you are literally the GOAT , thankyou sir , your 3 hrs of class goes like "woosh" ,
thankyou for uploading these lecutures and making our lives easier,
1:15:50 wait i don't understand why theta stops at pi/2
+Professor Leonard please keep on going, can you do Calculus 4 next semester?
???
Calc 4... LOL!
buddydog1956 Calc 843283428 is real analyisis, thats the good stuff
Do analysis PLEASE
Mr leonard why region in 1:49:21 is not going from Q=π÷2 to -π÷2
professor, you are the Best in Cal3
2:47:30 Going 0 to pi seems to work anyway? We are still calculating volume after all, it's just that height is 1.
Can someone help please? Leonard says at 1:48:30 to use 0 to pi, how ever at 2:47:30 he says not too, and instead do 0 to pi/2 multiplied by two. Why is there a difference?
I have the same question bud
hello
I do not understand why at 1:46:42 theta is from 0 to pi?
I think it should be from -pi/2 to pi/2
copied from: @learningleopard996
Ignore this. It's for myself but use it if you find it useful. :D
Intro to double integrals over polar regions~ 0:00
Volume over polar region examples~ 22:45
Ex. 1~ 23:55
Ex. 2~ 42:15
Ex. 3~ 53:50
Ex. 4~ 1:03:00
Ex. 5~ 1:23:41
Side note about theta not always being from 0 to 2pi~ 1:34:51
Example of polar region where theta is not from 0 to 2pi~ 1:38:00
Explanation of how to determine theta bounds carefully~ 1:46:10
Volume between two surfaces~ 1:59:04
How area can be represented by volume~ 2:37:17
Ex. 1- (Area of region bounded by one polar function)~ 2:42:17 (choosing theta bounds~ 2:47:16)
Ex. 2- (Area of region bounded by two polar functions)~ 2:53:28
More volumes in polar form~ 3:01:03 (choosing theta bounds~ 3:07:59)
Volume using polar coordinates on a rectangular region~ 3:16:19 (used when function (integrand) inside is difficult)
I just wish I watched ur videos earlier not 4 days prior to my finals :(
2 days before the exam and I still have so many videos left. I’m watching them at 2X speed.
Thank you so much for your dedication to quality education videos. Your approach to teaching helped open the door for me to go back to school and succeed. I cannot thank you enough good sir.
1:06:38 how do you identify the region??? It is not clear in the question that you ask