The Fundamental Theorem of Calculus. Part 2 

just figured out that my college class makes this alot harder than it should be. just subbed

Did people stop taking Calculus because part 1 had like 700k plus

Every one of your videos makes calc very straight forward and easy to understand, thank you so much.

im literally watching all your videos because my professor sucks , thanks man

None of your videos have any hate comments. This is the type of work anybody needs

Thank you soooooo much for all your videos! Your videos helped me from high school calc to college calc, you're the real MVP.

Dude right now I am working full time and taking calculus in the summer. There is no way i would have passed this class without your help because my teacher does not go over a topic more than once. However, you sir make your videos simple and straight to the point. Thank you!

you are my savior, your video clip is 10x better than sitting in the classroom for 2 hours, so i clicked on the ad in your video everytime i watch it as my support...

Have my Calculus final in about 2 hours, watching your videos has seriously made doing homework and simply just understanding the material much easier. Thank you so much!

Very clean job, as always, Patrick. Thank you!

Your amazing, I'm able to understand you so well! Keep posting, thank you!

I've been watching a ton of these all day to help study for my midterm tomorrow and they have helped SO MUCH! I just wanted to thank you, because these videos are amazing and so helpful and I feel like I know the theorems so much better now that I've watched these. So thank you! :)

@Machammerballs well, if you add the arbitrary constant, it will always immediately cancel out when you do the subtraction, so when calculating a definite integral, people just leave it off because of this. there is no harm in using it though, of course (try it once to see!)

Wow. Thank you so much. You pretty much summed up ten pages from the textbook into this one video!

What the heck?! I was so lost in the lecture. Thank you so much.

Five Finger Death Punch commercial before calc. I like it.

oh my god you are absolutely awesome!

this and part 1 helped me more than any lecture or text book ever could, you're awesome man thank you so much.

Thanks to you I actually understand calculus sometimes! Thank you for all the effort that you put in!

short, sweet, and simple. And that is why i love you.

you save my life in calculus, thank you!

Thanks for all the help! Everything is so much more easier when u explain!!

Hey Patrick! Thanks for the help. This lesson probably didn't mean much to you, but it was very helpful to me.

My teacher ATTEMPTED to teach us this for over an hour... This brilliant man explains it perfectly in less than 4 minutes... THANK YOU!!!!!

On a scale of 1-10, you're FREAKING AWESOME. Thanks a ton!

Nice explanation sir ... Thank you so much 👍

Instantly Subscribed and recommended to my entire AP calculus class.

THANK YOU SO MUCH!!! After going to class I felt so depressed... after watching this video I feel like I can do anything :D YOU ARE THE BEST!!!!!!

this is the simplest explanation to this date, 12 years later thank you

I like it so much.Thank you patrick. I thought math is difficult you made it so easy...It is amazing! Now I know......

Nice and simple, straight to the point!

you are the best math teacher I have ever met, thank you so much!!

your videos are going to help me get through Calc II this fall thank you!!!

spent an hour crying just for you to explain it in 3 minutes bless u

@sgilbret1 8 is correct. not sure where 12 is coming from.

my calc book has an entire section devoted just to explain that... 10 page later.. I was totally confused, 3 minutes after that your video saved me .. thANKS !!

God bless you Patrick...

thank you so much. Midterms are coming up and you are saving my life.

i wish you were my professor You make it so easy for me to understand thanks

lol all those hours in AP Calculus and it took me 5 mins to understand it here..Thanks bro!

Patrick, your a fucking genius. It takes my profs 2 hours to teach every section, and its so hard to understand. But, you teach it in under 10 mins perfectly and I get everything. :D

So true! I had the exact same sentiments going on.

@belittle12 :) just trying to keep it efficient

yes, the website is a pain to keep updated; many more videos on youtube

thank you so much. I finally understand this now

@xsinax15 the way to calculate an anti derivative is (X^n+1)/(n+1).. so basically you add 1 to the exponent of the term and divide by the new exponent

Hey Patrick! Your videos rock, but this one missing from your site! You have part 1 on it, but if part 2 isn't listed. Anyway, thank you for being the best math teacher I've ever had. :)

@NguyenEmpir3 add the constant and watch what happens

God bless you Patrick.

i wanna say this is a great video of basic idea of the fundamental theorem, but it will be great if you can do some more complicate examples, that will be really useful. you doing the good job!!!

You're the man.

SUPER SUPER SUPER 1MP0RTANT STUFF!!!! N0 J0KE MAN...N0 J0KE

@GKsArmy no, questions are ok, i mess up too :)

hey your video really is very helpful! thanks alot! it helped me alot more than my 140$ textbook....

Patrick, please make more ODE and Linear Algebra videos. I am gonna need those lol.

Thank jeebus for patrickjmt

THANK YOU I FINALLY UNDERSTAND THIS !!!!

Thank You so much!! I finally get it!!!!

please everyone support patrickjmt on Patreon, he truly deserves it!

I love you man, and I'm a dude. Don't take it the wrong way, it's just that a three minute video taught me more than an 80 minute lecture from my calculus professor. Keep it up, bro!

You are amazing.

A definite integral has a start point and an end point such as A to B. An indefinite point doesn't. I believe for an indefinite integral you just solve for the anti-derivative while on on a definite integral you solve for the anti-derivative with your B and A values plug in and subtracted in the form F(B)-F(A) where Capital F is the Anti-derivative.

thank you a lot

could you have done some more examples like in your FTC part 1 video?

Hey man your videos have helped me so much and I'm subscribed to your channel. I have a question though. In class, my teacher gave some sort of sliding car problem using integration. She gave us an initial velocity and its final position and then she asked us to find the position function using integration. She used like three different C (constant) values and she wrote it kind of funny. Do you have a video that explains this? Thanks for your help!

I'm pretty sure I could have skipped the last 16 weeks of calculus this semester for 1 day of watching your channel.

@pbezunartea Try that with more complicated functions. There are functions which have no anti derivative but do have an integral. In fact they actually mean two different things: the anti derivative is defined to be the opposite of taking the derivative, the integral is defined to be the area under the curve. It happens that in most cases those two are equal.

Do you have anything on the applications of integration... like finding the volume

thank you

Do you do math problems like everyday to get better? I don't know but I always get to study on the day before the exam :(

how do you know if an integral does not exist?

Hi Patrick, i have a quick question, when you take the antiderivative, you don't have to add a "C" to it?

Patrick, can you briefly tell me the difference b/w indefinite and definite intergeral? Thanks.

@pbezunartea This is one of the cases where the naming is a bit of a problem: Let's have the functions f(x) and F(x) such that f(a)-f(b) is the area under curve from a to b and F(x) is not differentiable. Then F(x) is the indefinite integral of f(x)dx however it is not its anti derivative because that would imply that taking the derivative of F(x) would give you f(x) back, which it doesn't since F(x) is not even differentiable. If it doesn't work in both ways it's not an anti derivative. :P

What's the difference between part one and part two?

come back any time :)

Why is there a dx in the notation? Why doesn't it cancel out the antiderivative?

Sorry, this is a dumb question, but why doesn't he add a constant C anywhere? I just started calc II yesterday, and the last thing we did last semester was antiderivates and I remember a couple times losing some points because I forgot to add the constant.

Thanks Man :D

@GammahooX "Try that with more complicated functions." I didn't use any functions, that was the general definition. "the anti derivative is defined to be the opposite of taking the derivative, the integral is defined to be the area under the curve" Well... the definite integral yes, it represents the area under the curve. Would the INdefinite integral then be the same as the anti derivative? Honestly I think it's just a matter of nomenclature. anti derivative = indefinite integration

You're a huge help!!! thank you! but what about for negative powers? would x^-3 be x^-2 or x^-4?

@pbezunartea Well they're not the same thing - it just turns out that the anti derivative can be calculated by using an integral.

Is the use of the term anti-derivative very spread? I remember using things like "integral" instead... :P

i learn more the day before the exam than the entire semester in university lmao

what happens if the bottom limit isnt 1

thanks a lot

@GammahooX "Well they're not the same thing" How are they different? If dF/dx=f(x), then the anti-derivative F=∫f(x)dx... how is that not the integration?

This is actually the main method they teach in college

What happens when the upper limit is infinite?

nah, i just chuck it out the window when i drive or i set it on fire in the backyard.

You'd make a fine professor. Most my math professors can't write, speak, or give 2 shits.

You, are a god.

Hello, sir. May I ask you why didn't you consider the constant C when you integral this function?

part 2 for me part one was anti derivatives

If I ever become a billionare I'll give you a million dollars or two for being so awesome :D

beautiful...

if you had a link from vid one, you would maybe have 150k views here

seconded!

your videos >>>>>>>>>>>>> any other calc video on youtube

in my country this is called barrow´s rule