##### The Riemann Sum!

Let's talk about the Riemann Sum!!

Okay, so the Riemann Sum does the same thing that an integral does, as it is essentially measuring the area under a curve! What this means for 𝛑, is that you can measure the area (well, approximate the are) of a semicircle, multiply it by 2 to get the area of a circle, and divide by r**2, to solve for an approximate value of 𝛑!

To imagine the Riemann Sum properly, think of a semicircle, with both ends touching the x axis. Now, imagine this:

This is a good visualization of integration (or more specifically, the Riemann Sum), but what is actually going on?

Well, there are two ways to explain this: The Calculus way, or the Summation way.

First off, the Calculus method is fairly concise and easy to read, but does not really help all that much:

Cool, right? (Note from after I took that photo, the integration is not approximately the area, it is exactly the area.)

Now, the Summation way (aka the Riemann Sum) makes all this so much clearer!

(In case you do not know how to read this one, it is basically saying that, as n → ∞, it sums all of the areas of all n rectangles that fit under the curve.)

Note that, when plugged into a calculator, you can say:

Please let me know if you have any further questions!!!
^ ^*

Oh, also, please pardon my poor penmanship.. T ~ T I have not written English too much lately. (Lol)

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DynamicSquid (4622)

I love the tutorial! Expect I didn't understand a thing... I'm trying to learn higher levels of calculus (like integrals and all that), and I even have a book on it, but... eh, I don't wanna do Math lol

LizFoster (648)

@DynamicSquid Well, if you have questions, I would gladly answer them! Calculus can be super cool and fun.

luffy223 (6)

if you wana see calculas visualy see three blue one brown chanal https://youtu.be/WUvTyaaNkzM @DynamicSquid

LizFoster (648)

@luffy223 I never got around to putting it in, but I meant to post his "Essence of Calculus" Series as a link on the post (Lol)

luffy223 (6)

the best chanal for math seekeris i am affriad of being involved of his diffrential calculas [email protected]

luffy223 (6)

numberphile is a good one [email protected]

LizFoster (648)

@luffy223 True, that channel has lots of good stuff on it
^__ ^*

luffy223 (6)

philosphy math and physics are my [email protected]

LizFoster (648)

@luffy223 I can tell! (Lol)

(Same here #_ _# )

What would the value of f^(n)(a) be? I am slightly puzzled as to how I would code that..

DynamicSquid (4622)

@LizFoster I have a quick question, where did you learn calculus?

LizFoster (648)

@DynamicSquid Since my school has yet to teach me, I had to learn it on my own time. T~T

I first really discovered and enjoyed the idea of calculus and integrals and all that stuff when I came across the graph of the equation e^x. It fascinated me that the area under any given x of this equation, is the equation!

I learned the bulk of it through video explanations and through the ever-helpful Mathematics Stack Exchange. If you'd like, I can supply you with some handy channels on YouTube for understanding the basics. ^__ ^

(Although, odds are, you probably know Calculus already _ _)

DynamicSquid (4622)

@LizFoster Well i just know the basics of it like limits and derivatives. Any links for the more advanced stuff?

LizFoster (648)

@DynamicSquid I would highly recommend one of my favorites:
blackpenredpen.
There is also James Cook, but I do not really like that he insists on beginning every lecture with a prayer (of course, that is just me, he is alright at explaining things otherwise).

luffy223 (6)

are you asking f to the power (n*a) [email protected]

LizFoster (648)

@luffy223 No, the Wikipedia page says that a Taylor series can be written in Sigma Notation, like so:

LizFoster (648)

@DynamicSquid Oh, and for blackpenredpen, obviously not all of his videos are Calculus-themed, but he probably has a playlist of them.

luffy223 (6)

as for learning culculas i highly recomend CALCULUS, FOURTH EDITION by robert smith and roland for all bacics of calculas untill second order diddrential equstion
if any one need it i have it it is from where i am studing from now @LizFoster

luffy223 (6)

I relly do not know but certinly you should try finding a code to find higher drivtion at first for any [email protected]

DynamicSquid (4622)

@LizFoster Cool! I'll definitely check them out. Thanks!

LizFoster (648)

@DynamicSquid No problem! As always, if you have any questions, I'd happily help you out!