Lol. He says "Stop me if I make a mistake", Like he ever makes mistakes. This guy is single handedly helping me pass my Calculus 2 for Engineers class in college right now. thank you so much.
He has made mistakes in several videos - he just usually catches them, or is corrected by one of the students. And this actually makes him more human, instead of those robot teachers who pretend like they never ever make mistakes.
This is such a strange lie... it's not possible to fail the first exam and end the class with a 97%. You'd have had to literally get 100% on everything else and the first test would have still had to have been only like 10% of your final grade.
0:00 Setup 14:40 Ex 1 21:40 Ex 2 38:50 Ex 3 42:00 Ex 4 51:50 Ex 5 1:00:19 Ex 6 (in terms of y AND x) - 1:07:45 What are you integrating w/respect to? The hard way. 1:27:23 The easy way
TIP: @36:00 It helps to separate the constants from the variables when you go to evaluate the integral. For the first term in the integral, begin by writing 1/2 x^2 instead of x^2/2. That way you can pull out the 1/2 and be left with evaluating 3^2 - (-2)^2 which is much easier. You get 9 - 4 which equals 5 and then you multiply by 1/2 to get 5/2. Every time you see a constant, get rid of it before plugging in your bounds to evaluate and you'll make the math a whole lot easier and save yourself from doing the same work over and over again. :)
Going to my calculus lectures i was only getting 60-70 on my quizzes, using your videos ive gone up to 90-95 on my quizzes! Thanks Professor Leonard, you're a life saver!
honestly i dont usually comment on videos but sincerely thank you!!! you are taking the time to make us fully understand concepts and don't rush it like professor in the Top 20 universities in the world, however they expect students to understand things on the fly and being a student in these type of universities, people just expect you to be smarter than everybody and do well on everything but honestly we struggle to understand like everybody and i feel that professors are not aiming to teach us anywhere more than a hunger game attitude of keeping an average and testing who can survive. Truthfully thank you
You should be getting some sort of teacher of the year award 🥇 your teaching methods are outstanding. Your students are lucky to have you. Thanks for taking your time to teach the rest of us
If there is anyone for the first time, I just want to tell you that Professor Leonard helped me to get A* in calculus easily. He has made things simpler for al of us. I am following his videos since 2016.
Dear Brofessor Leonard, I'm on summer break between Calc 1 and Calc 2 and I've been going through your lectures of both. It's wonderful that you never assume that your students know things. You've cleared up so many random areas of confusion about logarithms, exponent rules, trig/inverse-trig, properties of this or that thing, all while watching videos that weren't even covering that topic.
I would not have passed my CAL 1 final without you. I am so thankful you decided to make videos and post them on youtube. I love math, but lately I have been completely lost.
+Fine Fina This man is awesome . I wish we had a teacher like him , but some points i do not understand that which school is that . I'm studying in university but i do not know if you guys are from university or some other school types. Sorry if my english is not enuough by the way. I hope i wont disturb you and you will answer me
Jesus, I am so grateful for all your videos!! 🙏 So brilliantly explained! Don't stop teaching ♡ You are allowing us to painlessly understand, all these otherwise excruciatingly difficult classes.
He spends the exact amount of time as my professor, but the outcomes are absolutely different... so nicely explained, good pace not too slow nor too fast.
Oh my goodness...I am so glad I found this channel...I think you're going to be the reason I pass Calc 2. Thank you so much. You are my professor now lmao.
I am so grateful to you Professor Leonard! If there weren't your classes on RUclips I wouldn't be able to understand the materials professors are teaching in our university. You are being extremely helpful even to some people in Armenia.
*TIP:* @36:00 It helps to separate the constants from the variables when you go to evaluate the integral. For the first term in the integral, begin by writing 1/2 x^2 instead of x^2/2. That way you can pull out the 1/2 and be left with evaluating 3^2 - (-2)^2 which is much easier. You get 9 - 4 which equals 5 and then you multiply by 1/2 to get 5/2. Every time you see a constant, get rid of it before plugging in your bounds to evaluate and you'll make the math a whole lot easier and save yourself from doing the same work over and over again. :)
You've saved me yet again. You've helped me through community college when I was working full time and studying at night and you're helping me now that I've transferred to Stanford. Can you teach others to teach like you?
Thanks Professor Leo, I studied Statistic and Calculus 1 via Pro.Leo's lectures, and then I got A on these classes. Mr Leo, now I am taking Calculus 2 and I haven't not understood about work done, and the section "Application to physics and Engineering to find F and P and Moment. Pro Leo could you please post the lecture for these part in your channel. You are great Professor, and your lectures have helped me alots. Thanks so much
Ive watched all his videos so far for this playlist and not once has he lost me or confused me, this man has a true god-given gift for teaching. Thank you Professor Leonard
i honestly dont understand a lot of the stuff my teacher talks about in class, but when i watch his videos everything makes complete sense. I would not be passing calc without you!
love your methods of teaching you really helped me out in calc2, kept me at an A average for the class. Now I'm just rewatching videos to study for my finals.hope everything goes well
Thank you! You have the most amazing talents of making the very complicated subjects in calculus understandable. Your lectures help me peruse my dreams. Much love!
Love how you ask the class if they like racing before using it as an example. A lot of other teachers would be like this example is about knitting and I don't care if you like it or not 🤣
Everything explained very well, not just rushed to go ahead. Thanks Professor. I never had a good teacher since my school time that's why my base is very poor. You are amazing.
Superman coming to rescue the world... well, at least he's helping me greatly at the moment. Professor Leonard always simplifies complicated topics and does it interactively. Thank you for teaching effectively!!
Thank you so much Professor Leonard. Everything makes so much more sense thanks to your lectures and its entirely free. You are doing a great service to every student in America.
Hi, professor Leonard, from Ensenada, México, thank you very much. Im a gramma but I wanted to stydy calculus and found it a little complicated some times, you have been of great help.
Thanks god bump into this. Have my midterm this week and have to many thing that i forgot. Thanks Professor Leonard. Best regards from Malaysia Student :)
i'd love to meet professor leonard at a red light and race with him hahaha what a nice guy. thanks for all, see ya on calculus 2 playlist. greetings from chile by an autodidact
I've been having a really rough time getting rebooted on calc because of stress, and this is very helpful for understanding from the ground up. Just need to get muuuch better at basic conversions and trig identities...
My Calc 2 teacher at the university of Akron is probably the worst math teacher I have ever had I sat through 20 minutes of this lecture and understood more from this video than 2 hours in my actual lecture.
At 58:28 he says if the cars pass each other (or I guess if the velocities pass each other since that's what's in the graph), you'd have to make separate integrals for each interval. That's *NOT* true. That would be the case if we wanted to know just the abstract concept of the total area between those curves, but in order to get the final distance between the cars, net signed area is actually what we want. In the real world, it is true that if one car is faster than the other and then vice versa, the distance between them "cancels out", just like net signed area. If we actually did separate integrals and got the total area, we'd get the wrong answer. That wouldn't be the distance between the cars.
🎯 Key points for quick navigation: 00:00 *📐 Calculus 1 Lecture 5.1: Introduction to Finding Area Between Two Curves* - Introduces the concept of finding the area between two curves using integrals. - Explains that the area between two curves is determined by subtracting the integral of the lower curve from the integral of the upper curve over a defined interval. - Emphasizes that the resulting area is always positive if the upper curve is consistently above or equal to the lower curve over the interval. 03:09 *🧮 Understanding the Requirements for Finding Area Between Two Curves* - Requires the upper curve to be greater than or equal to the lower curve throughout the interval for a positive area. - Illustrates with graphical examples where the condition is met versus where it isn't. - Emphasizes the necessity of consistent positioning (upper or equal) of the curves for correct area calculations. 06:09 *📝 Application of Integrals for Finding Area Between Two Curves* - Demonstrates the application of definite integrals to find the area between two curves. - Shows how to combine integrals when both curves have the same bounds of integration. - Explains the process of setting up integrals for upper and lower functions and simplifying calculations accordingly. 25:13 *📈 Understanding quadratic equations and factoring* - Explains how to factor quadratic equations such as \( x^2 - x - 6 = 0 \). - Demonstrates the process of factoring and verifying solutions algebraically. - Highlights the importance of identifying solutions correctly in algebraic equations. 26:02 *📊 Determining bounds for integration based on intersections* - Discusses how to find the bounds of integration using intersections of graphs. - Emphasizes that intersections define the limits of integration for calculating areas. - Illustrates with examples how multiple intersections affect the number of integral calculations needed. 27:56 *🔍 Identifying which function is on top for integration* - Teaches methods to determine which function is on top within a given interval. - Emphasizes the role of function evaluations to establish dominance in integration setups. - Shows how choosing a test point within the interval helps in identifying the superior function for integration. 30:33 *📐 Setting up integrals to find area between curves* - Guides through the process of setting up integrals to calculate the area between two curves. - Demonstrates the integration setup based on which function is on top for each interval. - Highlights the importance of correctly setting up integrals using determined intervals and function dominance. 52:40 *🏎️ Velocity comparison and area between curves* - Explains the concept of comparing velocities over time and finding the area between two velocity curves. - Area between curves in this context represents the distance one car (me) is ahead of another (you). - Velocity curves allow for precise measurement of who wins a race and by how much, unlike simple velocity comparisons. 54:06 *📏 Calculating distance using area between curves* - Demonstrates how the area between two velocity curves corresponds to the cumulative distance covered by each car. - Integrating velocity curves yields distance traveled, providing a clear measure of how far one car is ahead of another. - This method offers a detailed comparison of positional differences over time, enhancing race analysis beyond simple speed comparisons. 01:00:24 *🔍 Integrating curves for area calculations* - Discusses the process of integrating curves with respect to the x-axis to find areas bounded by multiple functions. - Emphasizes the need to express functions in terms of x to correctly set up integrals for area calculations. - Points out the importance of identifying intersection points and understanding function behavior across different intervals for accurate integration. 01:17:40 *📐 Calculating area between curves* - Explanation of integrating to find the area between curves in calculus, - Importance of correctly identifying upper and lower functions for the integral, - Steps to set up and solve the integral for area calculation. 01:19:25 *📊 Simplification before integration* - Importance of simplifying expressions before integrating, - Tips on combining like terms and reducing complexity, - Demonstrating step-by-step integration and simplification. 01:27:18 *🔄 Changing variables for easier integration* - Explanation of changing variables to simplify integrals, - Demonstrating how to switch from x-axis to y-axis calculations, - Applying variable changes to specific calculus problems for clarity. Made with HARPA AI
Hi, I've watched all of your videos up through this one in order to study for the Calculus CLEP. You are a great instructor, and I learned a lot. However, your series is not quite enough to pass the CLEP. On the test there were a lot of questions about ln and e, as well as particle motion/velocity. It would be great if you could connect those dots on your Calc 1 playlist. For now I am going to study through your Calc 2 playlist and take the CLEP again in 3 months. Overall though, great great videos!
Just a heads up at 1:20:28, you should have brackets for integral from 0 to 1 of sqrt(x)-(x-2)dx. By the current notion, you are implying that the integral from 0 to 1 applies only to the sqrt(x), not the rest of the -(x-2)
if i am not wrong, he was explaining the Rolle’s Theorem when he said when f is continuous over the closed intervals [a,b] and differentiate over open intervals (a,b), then there must exist a number c when f’(k)=0.
At 6:30 the area between 2 curves is the difference between the 2 integrals of the respective functions. Why would not the C get involved in this case, since the C value is different for these 2 functions?
Lol. He says "Stop me if I make a mistake", Like he ever makes mistakes. This guy is single handedly helping me pass my Calculus 2 for Engineers class in college right now. thank you so much.
same fam
He has made mistakes in several videos - he just usually catches them, or is corrected by one of the students.
And this actually makes him more human, instead of those robot teachers who pretend like they never ever make mistakes.
Hes not a god for god sake every human makes mistakes...?
Calculus for Engineers II class. Struggle-cuddling these videos rn while I try not to go rock back and forth in the corner of my room lmao
Yep, waaay better than my actual prof that just put a 5 min video explaining the whole chapter -_-
Legit was failing my first exam.... this man brought me up 70% on exams. I ended my CALC 2 class with a 97%. Love this man so much lol.
Nice, well done!
that's not possible so ur lying
@@lcv8525Unless the course doesn’t care much about test scores…unlikely.
its also possible for professors to drop the lowest test score. Some of them do that.@@lcv8525
This is such a strange lie... it's not possible to fail the first exam and end the class with a 97%. You'd have had to literally get 100% on everything else and the first test would have still had to have been only like 10% of your final grade.
0:00 Setup
14:40 Ex 1
21:40 Ex 2
38:50 Ex 3
42:00 Ex 4
51:50 Ex 5
1:00:19 Ex 6 (in terms of y AND x)
- 1:07:45 What are you integrating w/respect to? The hard way.
1:27:23 The easy way
Thank you
Hero
*TIP:* In Calculus 3, these concepts will be extended to find the area between 3-D surfaces and 3-D objects.
TIP: @36:00 It helps to separate the constants from the variables when you go to evaluate the integral. For the first term in the integral, begin by writing 1/2 x^2 instead of x^2/2. That way you can pull out the 1/2 and be left with evaluating 3^2 - (-2)^2 which is much easier. You get 9 - 4 which equals 5 and then you multiply by 1/2 to get 5/2. Every time you see a constant, get rid of it before plugging in your bounds to evaluate and you'll make the math a whole lot easier and save yourself from doing the same work over and over again. :)
@@taekwondotime wow
Going to my calculus lectures i was only getting 60-70 on my quizzes, using your videos ive gone up to 90-95 on my quizzes! Thanks Professor Leonard, you're a life saver!
I DONT GO TO CLASS FOR CALC 1 OR CALC 2 & PASSED 97% BECAUSE OF YOU NEVER HAD TO GO TO CLASS THANK YOU NO ONE ELSE EXPLAINS IT LIKE YOU !!
This is the best thing that has ever happened to youtube!!!!
I agree
Really true
honestly i dont usually comment on videos but sincerely thank you!!! you are taking the time to make us fully understand concepts and don't rush it like professor in the Top 20 universities in the world, however they expect students to understand things on the fly and being a student in these type of universities, people just expect you to be smarter than everybody and do well on everything but honestly we struggle to understand like everybody and i feel that professors are not aiming to teach us anywhere more than a hunger game attitude of keeping an average and testing who can survive. Truthfully thank you
i'm in UofT and prof is the worst.. tbh I would fail without youtube. online learning sucks
You should be getting some sort of teacher of the year award 🥇 your teaching methods are outstanding. Your students are lucky to have you. Thanks for taking your time to teach the rest of us
If there is anyone for the first time, I just want to tell you that Professor Leonard helped me to get A* in calculus easily. He has made things simpler for al of us. I am following his videos since 2016.
Because of you today, I was able to transfer from Merced College to Cal Tech, your videos absolutely helped me get transferred to caltech
Perfect example of true teacher. Your way of teaching exceeded my expectations.
Quarantine classes need you, I'm so glad I found your page!
Dear Brofessor Leonard,
I'm on summer break between Calc 1 and Calc 2 and I've been going through your lectures of both. It's wonderful that you never assume that your students know things. You've cleared up so many random areas of confusion about logarithms, exponent rules, trig/inverse-trig, properties of this or that thing, all while watching videos that weren't even covering that topic.
Nice voice, clear handwriting, pleasant demeanor, concise notes and lectures. I wish I could have Prof. Leonard for all of my math classes.
I would not have passed my CAL 1 final without you. I am so thankful you decided to make videos and post them on youtube. I love math, but lately I have been completely lost.
I find myself raising my hand when I watch your videos. lol
+Fine Fina This man is awesome . I wish we had a teacher like him , but some points i do not understand that which school is that . I'm studying in university but i do not know if you guys are from university or some other school types. Sorry if my english is not enuough by the way. I hope i wont disturb you and you will answer me
Same :V
me too
Same here🙌
Same here
My professor is proficient in giving us assignments/tests/quizzes we are not prepared for.
Thank you for doing these videos
Jesus, I am so grateful for all your videos!! 🙏 So brilliantly explained! Don't stop teaching ♡ You are allowing us to painlessly understand, all these otherwise excruciatingly difficult classes.
He spends the exact amount of time as my professor, but the outcomes are absolutely different...
so nicely explained, good pace not too slow nor too fast.
Oh my goodness...I am so glad I found this channel...I think you're going to be the reason I pass Calc 2. Thank you so much. You are my professor now lmao.
This is the most chad calc professor I have ever seen
Every professors should teach like you. Your teaching is simply amazing.
I am so grateful to you Professor Leonard! If there weren't your classes on RUclips I wouldn't be able to understand the materials professors are teaching in our university. You are being extremely helpful even to some people in Armenia.
hi...i wanna talk to you
I just found this channel and holy smokes this guy is good. I love how he goes into detail and asks questions that other youtube channels don't
At 14:10 you can hear someone saying really soft "it doesn't make sense" lmao i feel you bro
He said the opposite actually, "it makes sense"
I did'nt know that I am capable of doing calc until I stumbled upon your videos. Thank You!
my final is tomorrow and this helped me understand area between two curves so much better!
*TIP:* @36:00 It helps to separate the constants from the variables when you go to evaluate the integral. For the first term in the integral, begin by writing 1/2 x^2 instead of x^2/2. That way you can pull out the 1/2 and be left with evaluating 3^2 - (-2)^2 which is much easier. You get 9 - 4 which equals 5 and then you multiply by 1/2 to get 5/2. Every time you see a constant, get rid of it before plugging in your bounds to evaluate and you'll make the math a whole lot easier and save yourself from doing the same work over and over again. :)
I almost lost the semester if it weren't for your videos. They saved my life 😢
You've saved me yet again. You've helped me through community college when I was working full time and studying at night and you're helping me now that I've transferred to Stanford. Can you teach others to teach like you?
Thanks Professor Leo, I studied Statistic and Calculus 1 via Pro.Leo's lectures, and then I got A on these classes.
Mr Leo, now I am taking Calculus 2 and I haven't not understood about work done, and the section "Application to physics and Engineering to find F and P and Moment. Pro Leo could you please post the lecture for these part in your channel. You are great Professor, and your lectures have helped me alots. Thanks so much
Ive watched all his videos so far for this playlist and not once has he lost me or confused me, this man has a true god-given gift for teaching. Thank you Professor Leonard
i honestly dont understand a lot of the stuff my teacher talks about in class, but when i watch his videos everything makes complete sense. I would not be passing calc without you!
honestly i dont even waste time watching the lecture videos coz i know ill end up here. so i just come to you and you never disappoint.
big thank you
love your methods of teaching you really helped me out in calc2, kept me at an A average for the class. Now I'm just rewatching videos to study for my finals.hope everything goes well
wow, youre a lot better than my college professor! Let's see if I can pass my final with your videos.
Did you pass !!?
did you?
Don't leave us hanging
Cmon, we want to know.
@@Windcutter_Flora lmao
You’re an excellent professor 👍! , your table helps me so much when doing these questions!
You help me got an A on my exam!!!! Thank you very much!!!!! I wish you could be my teacher at Long Beach City College.
Professor Leonard is amazing. What an awesome environment for learning. Great teaching!!!
Thank you! You have the most amazing talents of making the very complicated subjects in calculus understandable. Your lectures help me peruse my dreams. Much love!
Wow ! Your discussion is more understandable than my professor. Thank you very much Sir! :)
Your charisma and way of teaching is perfect. Thank you so much!
Love how you ask the class if they like racing before using it as an example. A lot of other teachers would be like this example is about knitting and I don't care if you like it or not 🤣
Thank you so much for helping students all around the world. It's been 6 yrs ago but I'm still watching them and your ideas feel so fresh
Words cannot describe how much you've helped me. Thank you very much.
Everything explained very well, not just rushed to go ahead. Thanks Professor.
I never had a good teacher since my school time that's why my base is very poor. You are amazing.
Superman coming to rescue the world... well, at least he's helping me greatly at the moment. Professor Leonard always simplifies complicated topics and does it interactively. Thank you for teaching effectively!!
I owe this man so much. If I could be any super hero I'd be Professor Leonard.
Thank you so much Professor Leonard. Everything makes so much more sense thanks to your lectures and its entirely free. You are doing a great service to every student in America.
you explain concepts with such lucidity, its refreshing
Hi, professor Leonard, from Ensenada, México, thank you very much. Im a gramma but I wanted to stydy calculus and found it a little complicated some times, you have been of great help.
Any amount of thanks would't be enough...I appreciate all of the hard work that you have put in.
I have been going in the lecture room for this calculus coming out empty but when I have watched this tutorial video i have known thanks professor
Thanks god bump into this. Have my midterm this week and have to many thing that i forgot. Thanks Professor Leonard. Best regards from Malaysia Student :)
man ive been watching you since last year when i wasnt sure which major to pick now im a sophomore in computer science
i'd love to meet professor leonard at a red light and race with him hahaha what a nice guy. thanks for all, see ya on calculus 2 playlist. greetings from chile by an autodidact
Im studying environmental engineering
And im pretty much failing my calculus class
Cramming up your video before my finals in 10 days
i really hate maths but after seeing your video i kinda get interested in it, i wish i had a cool lecturer like you in every module i do
I love your classes without you I would be lost. A+ here I come.
A month into calculus 2 at my school and I still haven't even graduated Professor Leanards calc 1! His students must be crushing calc 2.
I don't understand my teachers lectures so here I am and now I feel like my brain is expanding
I've been having a really rough time getting rebooted on calc because of stress, and this is very helpful for understanding from the ground up. Just need to get muuuch better at basic conversions and trig identities...
thank you dr , the extra good news about your lessons is that you take examples from the same book I have in my cal2 course .
book name plz
you're a REAAAAALLY GOOD teacher
proud to be a student of professor leonord
0:46:00, IMO, if x^3-x is an odd function, the net signed area is 0, and total area is 1/2.
Awesome job as always dude! Every single video is a banger🔥🔥🔥
i want this guy to be my cal professor lol great videos Mr. Leonard!
U R the Best 👌👌👌👌👌👌👌
Hands down impeccable
hey,you are very good in this. I'm astonished
Thank you Professor Leonard. Your videos are the best.
this is the most helpful video i've ever seen...Thanks professor 💖💖
My Calc 2 teacher at the university of Akron is probably the worst math teacher I have ever had I sat through 20 minutes of this lecture and understood more from this video than 2 hours in my actual lecture.
He just saved my life.WHERE THE HECK DOES HE TEACH ANYWAY?
+Aung Lin Brazzers.
+Aung Lin Merced College in California
+bell hill 😂😂😂😈😈😆😆😆😆😅😅
Now i guess it's North Idaho university
At 58:28 he says if the cars pass each other (or I guess if the velocities pass each other since that's what's in the graph), you'd have to make separate integrals for each interval. That's *NOT* true. That would be the case if we wanted to know just the abstract concept of the total area between those curves, but in order to get the final distance between the cars, net signed area is actually what we want. In the real world, it is true that if one car is faster than the other and then vice versa, the distance between them "cancels out", just like net signed area. If we actually did separate integrals and got the total area, we'd get the wrong answer. That wouldn't be the distance between the cars.
🎯 Key points for quick navigation:
00:00 *📐 Calculus 1 Lecture 5.1: Introduction to Finding Area Between Two Curves*
- Introduces the concept of finding the area between two curves using integrals.
- Explains that the area between two curves is determined by subtracting the integral of the lower curve from the integral of the upper curve over a defined interval.
- Emphasizes that the resulting area is always positive if the upper curve is consistently above or equal to the lower curve over the interval.
03:09 *🧮 Understanding the Requirements for Finding Area Between Two Curves*
- Requires the upper curve to be greater than or equal to the lower curve throughout the interval for a positive area.
- Illustrates with graphical examples where the condition is met versus where it isn't.
- Emphasizes the necessity of consistent positioning (upper or equal) of the curves for correct area calculations.
06:09 *📝 Application of Integrals for Finding Area Between Two Curves*
- Demonstrates the application of definite integrals to find the area between two curves.
- Shows how to combine integrals when both curves have the same bounds of integration.
- Explains the process of setting up integrals for upper and lower functions and simplifying calculations accordingly.
25:13 *📈 Understanding quadratic equations and factoring*
- Explains how to factor quadratic equations such as \( x^2 - x - 6 = 0 \).
- Demonstrates the process of factoring and verifying solutions algebraically.
- Highlights the importance of identifying solutions correctly in algebraic equations.
26:02 *📊 Determining bounds for integration based on intersections*
- Discusses how to find the bounds of integration using intersections of graphs.
- Emphasizes that intersections define the limits of integration for calculating areas.
- Illustrates with examples how multiple intersections affect the number of integral calculations needed.
27:56 *🔍 Identifying which function is on top for integration*
- Teaches methods to determine which function is on top within a given interval.
- Emphasizes the role of function evaluations to establish dominance in integration setups.
- Shows how choosing a test point within the interval helps in identifying the superior function for integration.
30:33 *📐 Setting up integrals to find area between curves*
- Guides through the process of setting up integrals to calculate the area between two curves.
- Demonstrates the integration setup based on which function is on top for each interval.
- Highlights the importance of correctly setting up integrals using determined intervals and function dominance.
52:40 *🏎️ Velocity comparison and area between curves*
- Explains the concept of comparing velocities over time and finding the area between two velocity curves.
- Area between curves in this context represents the distance one car (me) is ahead of another (you).
- Velocity curves allow for precise measurement of who wins a race and by how much, unlike simple velocity comparisons.
54:06 *📏 Calculating distance using area between curves*
- Demonstrates how the area between two velocity curves corresponds to the cumulative distance covered by each car.
- Integrating velocity curves yields distance traveled, providing a clear measure of how far one car is ahead of another.
- This method offers a detailed comparison of positional differences over time, enhancing race analysis beyond simple speed comparisons.
01:00:24 *🔍 Integrating curves for area calculations*
- Discusses the process of integrating curves with respect to the x-axis to find areas bounded by multiple functions.
- Emphasizes the need to express functions in terms of x to correctly set up integrals for area calculations.
- Points out the importance of identifying intersection points and understanding function behavior across different intervals for accurate integration.
01:17:40 *📐 Calculating area between curves*
- Explanation of integrating to find the area between curves in calculus,
- Importance of correctly identifying upper and lower functions for the integral,
- Steps to set up and solve the integral for area calculation.
01:19:25 *📊 Simplification before integration*
- Importance of simplifying expressions before integrating,
- Tips on combining like terms and reducing complexity,
- Demonstrating step-by-step integration and simplification.
01:27:18 *🔄 Changing variables for easier integration*
- Explanation of changing variables to simplify integrals,
- Demonstrating how to switch from x-axis to y-axis calculations,
- Applying variable changes to specific calculus problems for clarity.
Made with HARPA AI
Calculus final is in 3 days and you give me hope good sir
Awesome explanation. Thank you!
My teacher sounds like the love child of Squirdward and Sandy from the tv show Spongebob, you sir have been voted as my teacher from now on.
OMG , you cleared my concept ❤️
@10:50....this idea...damn I love math. I was literally smiling as he revealed this
Hi, I've watched all of your videos up through this one in order to study for the Calculus CLEP. You are a great instructor, and I learned a lot. However, your series is not quite enough to pass the CLEP. On the test there were a lot of questions about ln and e, as well as particle motion/velocity. It would be great if you could connect those dots on your Calc 1 playlist. For now I am going to study through your Calc 2 playlist and take the CLEP again in 3 months. Overall though, great great videos!
Very well explained. Prof. you really made this simple
may God bless professor leonard these tutorial are the best .thank you for your tutorial professor you the best and love you.
This guy is so good at teaching
Just a heads up at 1:20:28, you should have brackets for integral from 0 to 1 of sqrt(x)-(x-2)dx. By the current notion, you are implying that the integral from 0 to 1 applies only to the sqrt(x), not the rest of the -(x-2)
Thank you professor! You're one of the greats!
1:12:35
Why doesn't he use y=(x-2) in the first equation
x=(x-2)^2
it gives the same value of co ordinates (4,2) (1,-1)
Me: took precal roughly 6 years ago.
Prof: "Can you solve it?... I sure hope so"
Me: sweats.
This is the second lecture where he mentioned his car! I really want to know what he drives. He already mentioned twin turbos :)
if i am not wrong, he was explaining the Rolle’s Theorem when he said when f is continuous over the closed intervals [a,b] and differentiate over open intervals (a,b), then there must exist a number c when f’(k)=0.
At 6:30 the area between 2 curves is the difference between the 2 integrals of the respective functions. Why would not the C get involved in this case, since the C value is different for these 2 functions?
Yes, I know. Thank you.
Professor Calc Kent! That's great! Mild Mannered Calc Kent to the rescue! Slippery Slope Me!!
you're a great teacher
Love your lectures...
thank you professor....i really enjoy your lecture
May god bless you!
Amazing lecture, perfect explanation thank you so much!
Wow that car example is pretty neat!!!