Question

Fancy way of doing integrals.

Answer 1

\[\int\limits_a^b [(x+dx)^2-x^2]\] looks kind of like nonsense. But hear me out. This is just a telescoping series.

\[\int\limits_a^b [(x+dx)^2-x^2] = (b+dx)^2-a^2\]

Ok after you're good there, the rest is pretty obvious. Ok GL!

Answer 2

Any questions/comments, does any of this seem too ridiculous to be true?

Answer 3

I don't get it

Answer 4

\[\int\limits_a^b [(x+dx)^2-x^2]=[(a+dx)^2-a^2]+[(a+2dx)^2-(a+dx)^2]+...+[b^2-(b-dx)^2]+[(b+dx)^2-b^2]\]

Answer 5

I got it, I just wanted you to do more work.

Answer 6

It's like this: \[\sum_a^b [(x+1)^2-x^2]=[(a+1)^2-a^2]+[(a+2)^2-(a+1)^2]+...+[b^2-(b-1)^2]+[(b+1)^2-b^2]\]

Answer 7

You know how much I like work. =)

Answer 8

Your turn.

Answer 9

\[\huge W = \int\limits F(x) dx => \int\limits (Kai )dx\]

Answer 10

how do u type so huge

Answer 11

Because I'm batman, naw just put /huge

Answer 12

\[\huge hey \what \up\]

Answer 13

\[\huge OH\ SNAP\]

Answer 14

quick pretend to do integration

Answer 15

\[\int\limits_{-\infty}^\infty e^{-x^2}dx\]

Answer 16

\[\huge impossible...\]

Answer 17

\[\sqrt{(\int\limits_{-\infty}^\infty e^{-x^2}dx)(\int\limits_{-\infty}^\infty e^{-y^2}dy)}=\sqrt{\int\limits_{-\infty}^\infty \int\limits_{-\infty}^\infty e^{-x^2-y^2}dxdy}\]

Answer 18

\[\sqrt{\int\limits_{-\infty}^\infty \int\limits_{-\infty}^\infty e^{-x^2-y^2}dxdy}=\sqrt{\int\limits_0^{2\pi} \int\limits_{0}^\infty r e^{-r^2}drd \theta }\]

Answer 19

etc etc

Answer 20

Nice! Is it correct haha?

Answer 21

Yeah of course

Answer 22

I leave that as an exercise to the reader.

Answer 23

Haha, I was actually wondering about that for the past couple of days.

Answer 24

What else have you been wondering about lately?

Answer 25

A lot of things, none that are coming to my brain atm, but anyways it's nearly 5 am man, thanks haha. I'll ttyl, take care :P

Answer 26

Haha ok, well don't stress just curious, I usually think of really dumb things.

Answer 27

I actually thought that integral was impossible

Answer 28

It had been troubling me a for a while

Answer 29

I just asked a question in physics about how to find the moment of inertia for a 4D object. like srsly lol.

No integral is impossible, just power series lol.

Answer 30

4d object lol?

Answer 31

Yeah see, totally nonsense.

Answer 32

I'm too dumb for that

Answer 33

I'm too dumb for it.

Answer 34

I don't know anything about 4D stuff, so I thought I'd start asking questions lol

Answer 35

Mhm I wonder how you would draw 4d?

Answer 36

Is it even possible lol

Answer 37

I don't even know if I can draw 3d objects.

Answer 38

That's when the d starts making sense it's a whole new dimension waiting to be discovered.

Answer 39

non-integer dimensions? lol

Answer 40

Is it quantum mechanics that deals with " 11 dimensions."

Answer 41

Actually, if you have a non-integer base to your number system it turns a number like: \[1234+567\sqrt{10}\] if you write that number in base 10^(1/2) it turns out to be easily written as \[10253647\]

Answer 42

No, that's string theory.

Answer 43

Oh right, I'm too tired...>_>

Answer 44

I don't know anything else about string theory other than that nonsense with the -1/12 garbage.

Answer 45

Yeah I know nothing, I need to read more Michio Kaku books haha.

Answer 46

I've never read him. I think he's kinda a crackpot tbh. But that's just my impression of him. And I really just don't care for Neil DGT either. I don't know why, I just don't lol. Carl Sagan seems cooler of a dude.

Answer 47

TELL ME YOU JUST SAID YOU DON'T CARE ABOUT NDT

Answer 48

I love NDT haha, have you been watching the new cosmos?

Answer 49

I like the idea of mapping the entire number line to between -pi/2 and pi/2 by using arctan. lol

Answer 50

I saw the first episode. i don't have a TV so not really. It seemed alright, but I'm just not really into science without integrals lol

Answer 51

Ooh that's cool

Answer 52

Ah, all good, I'm always watching tv man..I need to start paying more attention to math, I have all these questions, but no idea how to answer them, and sometimes those questions don't "belong" here. Or I guess, I'm just embarrassed.

Answer 53

like the other day I learned how to calculate time dilation in relativity. It's really pretty easy and straightforward. Like seriously, just make one claim that c is constant and do a thing with a right triangle. I could show you if you like. Takes a few seconds.

Answer 54

Nah, don't be, just ask them when they come up and tag me or wahtever I like to think about stuff lol.

Answer 55

Haha alright, and lets leave relativity for tomorrow, though I'm very interested in that! I'm off to bed, take care man!

Answer 56

k later.

Answer 57

Actually show me the claim that c is constant. I won't be able to sleep otherwise.

Answer 58

Well the claim that c is constant is a postulate of relativity. Why would anyone ever believe this or come up with this? Well when you solve the wave equation using maxwell's equations for electricity and magnetism what comes out of it is a wave that has a constant speed.

That's just weird. But if you just assume it's true and follow out the logical consequences of it, it turns out to be useful and true. We are able to calculate time dilation of the satellites responsible for GPS so they're accurate since they're going fast enough for relativistic effects to no longer be negligible.

Answer 59

Why is reality like that? Beats the pellet outta me, but what it does seem to be is like we have sort of a momentum towards the future. If we move, we're sort of taking our vector and pointing it out of the future and into a direction. So you actually age less if you're moving. Of course for us humans it's not very noticeable since we're not nearly going fast enough.

Answer 60

so here in this picture a person in a train turned on a laser and it hit the ceiling. The distance it travels is the height of the inside of the train. From 4th grade we know distance = rate time time, so if the distance and rate are constant, then time must be what is variable.

\[d=c \Delta t_0\]

to a person watching it looks like this though: So the distance the photon went by someone watching is the hypotenuse, and that's just \[c \Delta t _1\] and from our reference frame outside the train, if the train has a velocity v, it has gone \[v \Delta t_1\] so doing a little pythagorus stuff \[(c \Delta t_1)^2=(v \Delta t_1)^2+d^2\]