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)
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.
there is an ad starts like "are you searching youtube for math help? from 2008 videos reeaaaaallyy?" and I was like bitch this is professor leonard, 2008 or 3008 doesn't matter for him, ppl will talk about this legend for generations :D
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.
I pay money to take multi and just watching Professor Leonard's lectures for free online. This man deserves a portion of every university's tuition for teaching the entire nation's students upper division math.
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!
This professor is amazing.I impressed with his way of explanation. I clapped my hand whenever he explains things in simpler terms. I learn not only maths but also how to teach math artistically. Thanks so much professor for offering us for free.
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!
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!
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
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!
I am gonna download each of professor Leonard videos and store them for my children so if anything goes wrong with youtube in future my children don't have to suffer from maths because of some teacher at school.
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
High quality teaching! I like that you really make sure the students understand every aspect of the subject and not just plug in numbers! 3 and a half hours goodies!
@@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
Wow this is a super good lecture. I remember learning this stuff years and years ago before youtube and not being able to watch lectures online. You had 1 instructor the office and lectures they had and whatever textbooks you could get. This guy's awesome. I remember Cylindrical coordinate systems blowing my mind at one time but this guy really helps break down every step. It's refreshing to see such a well crafted lecture. Not sure why youtube thinks I need a calculus lecture but I really enjoyed the refer class and will gladly watch this and other lectures. Wish this was a thing 20 years ago.
after watching any video of Prof.L, I'm compelled to say: Wow! that was the best lecture on the topic. i need a prof.L for my Quantum mechanics class now.
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.
I never ever comment here, but words can barely express my feelings of gratitude. Brilliant, brilliant videos! Thank you so much for what you are doing. I wish there were more teachers like you. You are absolutely amazing.
My god, what an amazing teacher. No wonder I never understood a single thing about calculus from my teachers!. Hope I can become as good at explaining as you are (I'm teaching programming at first year university courses).
@@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)
Freedom of information is fucking fantastic. This marks a primary shift in the delivery of education. My lecturer is awful - so here I am watching someone on youtube who I completely get. I feel sorry for those who had to teach themselves this from text books.
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.
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 ^^
thanks for being my professor during covid when my calc 3 professor doesnt hold any lectures and points us to you and khan academy for learning :) it really is a blessing
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.
I honestly don't know why I'm watching this, I'm majoring in computer science I have nothing to do with this but his way of explaining grabbed my attention tbh. Good luck to anyone who takes calculus 🙏🙏
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!!
I am here just to tell that i love u Professor and U r real Professor. Not like our professors who have degree but do not know how to teach. U are real Superhero and Livesaver of many students. Once more love u so much Professor Leonard.
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.
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?
the coolest professor. thank you so much. i will never get bored in your class. you release energy that makes things really interesting. wish I had cal classes with you
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 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.
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'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.
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.
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
This "Super Man" has a super brain that he can explain D.I. so so so detailed !!! Amazing, and he had upload all of this amazing lecture online free!, This super hero saved my life.
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.
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.
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))
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)?? 🤔😕
I liked the Pirate Joke ...Professor Leonard has helped me through my Associate Degree and my UnderGrad ...So thank you for your Commitment in ensuring we get this Complex theorems ..and if anyone has information on where he teaches i would like to enroll there for my Post Grad Studies ...feel free to hit me up
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
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'm a bit confused about how you find the range of theta. For ex. volume below 3x+4y+z=12, bound by region between x^2+y^2=2x and above xy-plan, the range of theta is 0 to pi But for the example of r=3costheta, the range of theta is 0 to pi/2. Why the range is not pi for this? Why it is 2*integral theta from 0 to pi/2?
The units at 2:41:00 are actually not different at all, because if you divide by a height that has the value 1 then you are actually dividing by 1 _unit of length_ - so you will divide out one of the dimensions, and thus end up with "square units of length".
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.
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
L'n'I'll ô.o'l''l'
O'
O'
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.
there is an ad starts like "are you searching youtube for math help? from 2008 videos reeaaaaallyy?" and I was like bitch this is professor leonard, 2008 or 3008 doesn't matter for him, ppl will talk about this legend for generations :D
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
I pay money to take multi and just watching Professor Leonard's lectures for free online. This man deserves a portion of every university's tuition for teaching the entire nation's students upper division math.
Not just the nation, the entire world.
Even here in India, even understandable of a 15 yr old like me.
T
AGREE!!!
Ubiobboiobjbjob Hhhhh ohjjjjjj I’m so excited about you guys so I don’t
Iiiiiio I uuu. 😊ao
I will literally set up a kickstarter campaign for Professor Leonard to upload Dif Eq videos...who's with me?! :D
I will be starting diff eq next semester... Lets make it happen!!
agreed!!
I'd chip in :)
make it so
You can support him on Patreon. He needs a two thousand dollars per month goal to make it happen -
www.patreon.com/ProfessorLeonard
Shoutout to everyone out there with me paying university tuition just to end up doing all of the best learning through this guy.
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 !!!
@@EmilioLopezFlores he prolly doesnt remember hahaha it was 3 years ago
Congrats🎉
This professor is amazing.I impressed with his way of explanation. I clapped my hand whenever he explains things in simpler terms. I learn not only maths but also how to teach math artistically. Thanks so much professor for offering us for free.
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!
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!
I swear, these lectures are good enough to allow you to pass an exam even without reading the course book.
Seriously.
i passed all my calc 3 exams by high grades by these lectures, amazing
@@mariagevorgyan1470 That's awesome.
exactly, what i said
got an 82 on my calc 3 test
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...
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!
I am gonna download each of professor Leonard videos and store them for my children so if anything goes wrong with youtube in future my children don't have to suffer from maths because of some teacher at school.
You would do better to remember this
That's smart AF
@@epistemophilicmetalhead9454 loooool
@@epistemophilicmetalhead9454 already did
@@SachinAnalyzes
I need to send this man flowers and a package of tight t-shirts. Saving my ass Cal I-Cal III. Much love
tight tee shirts ahahahah
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
High quality teaching! I like that you really make sure the students understand every aspect of the subject and not just plug in numbers! 3 and a half hours goodies!
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
This guy is like a nerdy jock and its awesome. He will beat you up and do your homework. keep it i^2 Prof.
Wow this is a super good lecture. I remember learning this stuff years and years ago before youtube and not being able to watch lectures online. You had 1 instructor the office and lectures they had and whatever textbooks you could get. This guy's awesome. I remember Cylindrical coordinate systems blowing my mind at one time but this guy really helps break down every step. It's refreshing to see such a well crafted lecture. Not sure why youtube thinks I need a calculus lecture but I really enjoyed the refer class and will gladly watch this and other lectures. Wish this was a thing 20 years ago.
after watching any video of Prof.L, I'm compelled to say: Wow! that was the best lecture on the topic. i need a prof.L for my Quantum mechanics class now.
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.
I never ever comment here, but words can barely express my feelings of gratitude. Brilliant, brilliant videos! Thank you so much for what you are doing. I wish there were more teachers like you. You are absolutely amazing.
My god, what an amazing teacher. No wonder I never understood a single thing about calculus from my teachers!. Hope I can become as good at explaining as you are (I'm teaching programming at first year university courses).
I wish every teacher and professor out there had your same passion! Thanks to you I have aced Calc 1-3! Thanks!
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)
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
hahaha
LMAO 🤣
watching the video on ritalin :D
You're the reason I've gotten A's my past two midterms. My tuition should honestly go to you!
Freedom of information is fucking fantastic. This marks a primary shift in the delivery of education. My lecturer is awful - so here I am watching someone on youtube who I completely get. I feel sorry for those who had to teach themselves this from text books.
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 RUclips 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.
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 ^^
thanks for being my professor during covid when my calc 3 professor doesnt hold any lectures and points us to you and khan academy for learning :) it really is a blessing
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.
I honestly don't know why I'm watching this, I'm majoring in computer science I have nothing to do with this but his way of explaining grabbed my attention tbh.
Good luck to anyone who takes calculus 🙏🙏
Thanks to you Professor Leonard, I improved greatly on my most recent exam after watching your Calc 3 videos
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!!
I am here just to tell that i love u Professor and U r real Professor. Not like our professors who have degree but do not know how to teach. U are real Superhero and Livesaver of many students. Once more love u so much Professor Leonard.
You are an absolutely jaw dropping genius of a teacher!
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.
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?
This guy has some real teaching talent, though the content was simple I saw complete video.
big thanks may god bless you and ur family
Holy moly wish I found this man earlier in my semester, so much makes so much sense now. 10/10 watch these long but amazing videos
Nearly DONE!!! Professor Leonard enriched my vacation! I love multivariable calculus!
the coolest professor. thank you so much. i will never get bored in your class. you release energy that makes things really interesting. wish I had cal classes with you
I appreciate how you placed your adds. Thank you for this smart move!
THANK YOU! Like honestly this kind of help and teaching should never be taken into granted.
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
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 ?
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.
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.
AAAAARRRRRRRRRRRRRRRRRRRRRRRDDDDDDDDDDDDDDDDRRRRRRRRRRRRRRRTHEEEEEEEEEEEEEEETA :d
These lecture videos are pure gold, thanks alot Professor Leonard
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.
not gonna lie, you're very handsome
Then you have a twisted understanding of beauty
What's wrong with you m8?
@@marbrydav9698 well he is good looking. Its just fact.
@@AndyU96 no u
No homo
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
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
This "Super Man" has a super brain that he can explain D.I. so so so detailed !!! Amazing, and he had upload all of this amazing lecture online free!, This super hero saved my life.
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.
Wow... I am having fun learning Calc from you. That's a first! haha
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.
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.
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))
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!!
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)?? 🤔😕
At 34:30, why would it not change to -π/2 to π/2?
Superman is teaching me calculus.
I fell asleep watching something completely unrelated and woke to this. Not mad. 10/10 will save this for later.
BEST PROFESSOR EVER. Thank you for all you do!
Thanks Prof Leonard, I am seriously ahead because of you.
Watch me crush the final exam on 5/27/2022.
Two questions:
In the example @1:33:42, I set my limits of r to 0
One of the most sensible videos on double integration
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?
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
east or west professor leanord is the best teacher
came here after being stuck in a problem but I knew where to came, you never disappoints thank you Prof
I liked the Pirate Joke ...Professor Leonard has helped me through my Associate Degree and my UnderGrad ...So thank you for your Commitment in ensuring we get this Complex theorems ..and if anyone has information on where he teaches i would like to enroll there for my Post Grad Studies ...feel free to hit me up
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
1:06:38 how do you identify the region??? It is not clear in the question that you ask
13:30
they're called Adderal.
You are AWESOME SAUCE!!! You make math GREAT AGAIN! Thank you for not being boring!
شكرا
thank you
gracias
Danke
grazie
merci
Please do differential equations Professor Leonard!
HANDS DOWN THE BEST CAL TEACHER!!!!
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'm a bit confused about how you find the range of theta. For ex. volume below 3x+4y+z=12, bound by region between x^2+y^2=2x and above xy-plan, the range of theta is 0 to pi But for the example of r=3costheta, the range of theta is 0 to pi/2. Why the range is not pi for this? Why it is 2*integral theta from 0 to pi/2?
There something upthere
Thank you so much Professor
Teachers like you made me fall in love with maths
Keep up the good work
For the example at 40:09, an alternate approach would be to rewrite 4cosθsinθ as 2sin(2θ). I feel like this
makes the u-sub easier
prof you have made my life easier.thanks oncemore
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
1:15:50 wait i don't understand why theta stops at pi/2
37:50 We don't need to change the bounds, because sinX*cosX equals (sin2X)/2. So the integral will be (-cos2X)/4.
yes. it can be done without u substitution.
i dont understand in example 1:47:14, theta can go from 0 to pi but not in example 2:47:00, can somebody expain to me
The units at 2:41:00 are actually not different at all, because if you divide by a height that has the value 1 then you are actually dividing by 1 _unit of length_ - so you will divide out one of the dimensions, and thus end up with "square units of length".
The example solved at 01:07:00, why is the upper region for integration? why not the lower part of the circle?
Mr leonard why region in 1:49:21 is not going from Q=π÷2 to -π÷2
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
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.
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.
professor, you are the Best in Cal3