Area Enclosed by a Polar Curve, Calculus 2

[@marcushendriksen8415]

"It's not because I cannot draw, but because it's not a circle." Since when were those two things mutually exclusive, huh? ;)

[@benjaminbrady2385]

Hey bprp. I've noticed that you seem more on edge towards negative comments lately. I hope that you're alright... more negative comments is a natural result of growth and most people aren't actually expressing hate but are often memeing or making an obscure reference. Remember, anyone who comments on more than one video is here because they love your videos, not because they hate them 1000000/1-x

[@blackpenredpen]

Thank you Ben.

[@hamiltonianpathondodecahed5236]

can someone please explain me that *100/1-x* thing

[@benjaminbrady2385]

@@hamiltonianpathondodecahed5236 1/1-x is the "Best friend" powerseries so by multiplying it, you're saying very best friend or best friends

[@blackpenredpen]

Benjamin Brady yes! : )))

[@BrainGainzOfficial]

Black pen red pen blue pen?! 🤯

[@nchanelle]

Brain Gainz haha I saw that!! 😂😂

[@blackpenredpen]

Hahaha

[@BrainGainzOfficial]

@Chanelle L Thanks!!

[@SiilverEdit]

the way you smile while explaining makes me happy and shows how much you enjoy keep up

[@blackpenredpen]

Thanks, will do!

[@frozenmoon998]

Art is not in painting, but in math, bprp! We do know your drawing was great.

[@blackpenredpen]

Awww thank you!!!!

[@joso5681]

"uploaded 36 seconds ago" "First comment 7 hours ago" Excuse me wot?

[@828burke]

Polar time coordinates

[@v6790]

Patrons get early access

[@S1nwar]

i like that you always simplyfy after writing it down. there is never a step lost

[@blackpenredpen]

S1nwar thanks

[@pronounjow]

You draw an amazing not-circle! It looks spot-on!

[@roderickwhitehead]

I don't believe I have ever seen integration by parts done that way... mind blown.

[@colly9888]

It's called the "tabular method for repeated integrals"! Very useful for trigonometric functions and e^x (:

[@uncagedstudios1811]

I got an A in Calc II this semester because of your integration videos. Thanks for your help.

[@kostantinos2297]

I'd just like to say thank you for your help though these videos, particularly on integration. I've been studying the topic on my own ahead of my lessons, and these videos are an awesome way to put the theory into practice and better comprehend what it's all about.

[@blackpenredpen]

Kostas T. My pleasure!!

[@김주한-n2h]

Hello l'm Korea student your math is great useful

[@michiplays00]

wow, perfect timing. Learning about polar area right now so thank you!

[@blackpenredpen]

Michi Plays I have more examples. You can check my playlist. : )

[@wolfieeeee256]

Wow this is actually really interesting :) Fantastic video!

[@blackpenredpen]

Joecake Gamedev thank you!

[@andres.robles6]

Haz un álbum de puros problemas de cálculo vectorial.

[@MoonLight-sw6pc]

Thank u blackpenredpenbluepen

[@blackpenredpen]

: )

[@eansengchang6840]

bro 7 hours how

[@sergioh5515]

@@eansengchang6840 I was wondering that too...but how🤔

[@blackpenredpen]

Sergio H it was released on my Twitter first last night.

[@r.d.x7403]

Just what I was looking for!

[@drpeyam]

Yaaaay! You’re the best!

[@blackpenredpen]

Dr Peyam thank you!! Btw, does your calc class cover this?

[@drpeyam]

blackpenredpen Sadly not 😣 We do polar coordinates, but not calculus with polar coordinates 😭

[@blackpenredpen]

Dr Peyam oh

[@k.n.vkishore8201]

You use colours of pens very nicely

[@slavii5772]

Can you do something about Diophantine equations?

[@General12th]

Find all the parametric solutions for (x,y,z)^3 = 31.

[@sergioh5515]

Very good review for calc 2...thnx bprp 😀

[@blackpenredpen]

Thanks!

[@CuriousNeon]

DI table caught me off-guard lol! Maybe I need to study or watch some videos to understand it more...

[@mothy4310]

Thank you for the education! Your videos were my motivation to foray into calculus and finally got me back into learning math on my own (doing math for fun lol) keep it up! :)

[@gabrielmedina2480]

Thank you!

[@kolinseward5033]

I don’t know what is going on in the universe but every time he posts it is somehow the same thing I’m going over the exact same time

[@blackpenredpen]

Because I know the secret! : )

[@blackpenredpen]

And I bet sequence and series are coming?

[@hamiltonianpathondodecahed5236]

he might be your student

[@chaoticoli09]

Great video. I am now interested in the sequence of areas of each loop.

[@blackpenredpen]

Hmm, it would be from n*pi to (n+1)pi and I wonder how the values will go.

[@chaoticoli09]

You can certainly write the sequence of areas as a function of n by using the bounds you've provided and I believe that you get 1/12 * pi^3 *(3n^2+3n+1)-pi/8. There may be some recursion you can develop from this, but I don't have any ideas off the top of my head. At least this allows you to provide the area of any loop that you want!

[@LifeIsBeautiful-ki9ky]

Explain about sketching a polar and parametric curves.

[@General12th]

So good!

[@blackpenredpen]

J.J. Shank thank you!!

[@mathmathician8250]

If the inner loop goes from 0 to π, then the outer loop goes from π to 2π.

[@hjk0150]

The area inside the outer polar curve and outside the inner curve should be 1/2π^3 if I did my calculations correctly.

[@antoniomodugno2753]

The larger circle area should be 7/12π^3-1/8π

[@hayzzzeus]

Thank you blackpenredpen for another amazing video. This will definitely help me out for the AP Calculus BC Test. Also could you please make a video on differentiating a polar function?

[@blackpenredpen]

I have a playlist in the description. : )

[@blue_blue-1]

No „rad“ included. Like it.

[@taba3514]

I think that the sign of the middle blue is not minus but plus(+½θ·sinθ = +(½)(¼)(π)=π∕8).

[@GabeTStarman]

Here’s an interesting question: how does one determine and prove mathematically whether or not the whole region 0 to 2π is similar to the region 0 to π?

[@blackpenredpen]

Gabriel Taylor Hmmm very interesting! My first thought is to compare how their arc length and area changes. But it’s just a quick thought. I wonder if anyone else has any other idea?

[@pierreabbat6157]

It's obviously not, since the graph has a cusp at θ=0, but at θ=π it passes through the origin with finite curvature.

[@StephenMolloyGoogle]

@@pierreabbat6157 Is that really true? For small angles, sinx=x, so x.sinx = x^2 --> finite curvature at 0. There shouldn't be any discontinuities anywhere since it can be differentiated as many times as you like. Another way to think about it is that the derivative of x.sinx will be x.cosx+sinx, which is clearly finite at x=0 (and all finite values of x). That also allows you to see that the gradient at the origin-crossings will change with a very interesting pattern -- 0, -pi, 2pi, -3pi, 4pi, etc.

[@StephenMolloyGoogle]

Actually, now I see that I goofed. Polar form. Duh. Ignore me.

[@ta_helado]

Start a linear alegra series

[@zackmercurys]

I wish I knew what is that D I table you always draw when you do integration by parts

[@flaviusclaudius7510]

I found this document on the subject; the first page is a proof, the rest showing how to use it (in table form): ramanujan.math.trinity.edu/rdaileda/teach/s18/m3357/parts.pdf

[@flaviusclaudius7510]

Also, pretty salty that I didn't come across this until the final year of my PhD >_>

[@blackpenredpen]

kzbin.info/www/bejne/o5LXdYlpjbFlgJo : )

[@user-sq5uq8mj5m]

Hi, blackpenredpen! What university did you graduate from?

[@BlokenArrow]

All of. Them

[@roderickwhitehead]

@@BlokenArrow - LOL

[@jarikosonen4079]

How can you write cartesian y=sin(x) in polar coordinates and then integrate the area? It in some sense has no area at all... but if plug to this equation it would give area, right? (Because curve isn't "closed"). This case in cartesian case it gives 0 area if n*2*pi integration range is used. This case same in polar could give area of circle with radius r=n*2*pi..(???)

[@pwootjuhs]

is it possible to compute when the area of a circular segment has the same area as a square with the same side length of the circles radius? I came to x-sinx=2. I don't see any way of getting any more out of this, but you might.

[@yarooborkowski5999]

Could You prove the integral form of area between curves in polar cordinates, please?

[@blackpenredpen]

Here's an example: kzbin.info/www/bejne/gKmse3-wedVggpY Again, the formula is just "the area of the sector" I will do the proof one day.

[@JamalAhmadMalik]

If it Area of the bigger circle looking thing, would it be \theta replaced by 2\theta?

[@marstruth1578]

Sign error: (I don’t have theta so using @) -2@ times -1/4 cos (2@) = 1/2 @cos(2@)

[@yoyoezzijr]

Theres a negative outside too

[@ВладиславБеляев-и1г]

is abs implicitly assumed there? sin(x) is negative for x=pi...2pi, so it is incorrect polar coordinates equation

[@iabervon]

No, r is allowed to be negative; it just puts the curve on the other side. That's why the second loop is also on top, despite the angle pointing down.

[@bhuvird178]

Good pa

[@joehughes9043]

What is that method of IBP?

[@justabunga1]

Joseph Hughes This is called the DI method. D stands for differentiation and I stands for integration. He is setting up the tabular method to show a quick way of doing this.

[@LiammTheCat]

Can you do a video talking about and introducing polar coordinates :) #yay

[@blackpenredpen]

I will put them in the description!

[@LiammTheCat]

@@blackpenredpen thank you! how about a video talking about polar curves vs rectangular curves?

[@lorahowsian6811]

In my calculator, I plug in the integral and I get 2.19, is this correct?

[@blackpenredpen]

Lora Howsian Hmmm try to enter what I got on the board to see if they match.

[@lorahowsian6811]

blackpenredpen I did, I got approximately 2.19, and i subtracted the result you got and I got 2.19 too.

[@thenewguy7527]

blackpenredpen #yaaaay

[@purim_sakamoto]

これも普通に積分しちゃっていいのね あれ、なんでそうなるんだっけ 基礎的な事が抜け落ちてるなあ

[@W.T.FISHAPPENING]

so easy...

[@therenaissance8322]

WHY DID YOU UNFOLLOW PAPA FLAMMY ON TWITTER?

[@blackpenredpen]

?

[@emperorpingusmathchannel5365]

Arya kills the night king. I am going to hell for this spoiler.

[@Mnemonic-X]

Does anybody know that the special theory of relativity is completely stupid theory that contradicts logic?

[@blue_blue-1]

Сергей Мишин, No, I didn‘t. Make a video.

[@Mnemonic-X]

@DY_Physics If they (Maxwell's equations) contradicts logic then they are fake. Is it right, isn't it?