integral of f is the area under the graph
but what is integral of xf(x) ?
The area under the graph of [math]x \longmapsto x\,f(x)[/math].
>>7998946
x^2f(x)/2 - ∫x^2f'(x)/2 dx
xF(x) - ∫F(x) dx
If f has a Taylor seriesz it's an equally comfty series
>>7998946
you need a dx nigga
>>7998946
The first raw moment is the mean.
It's the area under the curve of h(x) where h(x) = xf(x), moron.
>>7999024
Kek
>>7998946
the expected value of f(x)
>>7999065
/thread
Mass-Centre
>>7998946
Just Multiply the f(x) by x
Eg, f(x) = 5x, integral of xf(x) = int of 5x^2 = (5x^3)/3
Or let u = xf(x)
>>7998993
No, you really don't.
>>7999138
Yes, you really do.
>>7998946
Ever heard of partial integration ?
>>7998946
Y'all niggas don't know multuplication law
Int[xf(x)] = 1/2x^2(f(x)) × x*F(x)
>>7999321
It doesn't really serve a purpose.
>>7999387
It is part of the definition of an integral.
>>7999397
it's just notation, it's clear what he means. fuck off
>>7999412
No it isn't.
>>7998946
It is x*f(x) - Antiderivative(f(x))
>>7999412
No it's not. Without it the integral makes no sense.
Based on your ignorance of what the integral actually represents I'm guessing you're a retarded American. Jesus christ, I fucking hate Americans. All they learn is this mechanical bullshit, nothing related to actual math. Fuck America and fuck your retarded education system
>>7999387
It does as at the most basic level it shows what variable you have chosen to integrate with the respect to. For all you know, OP wants do integrate with respect to y and you now need to either pull of a substitution OR re-expess the integrand in terms of y. As for the fact you can 'guess' what he means, the point of mathematical notation is to make sure you NEVER have to guess.
>>7999412
>taking the sum of an infinite amount of noninfinitesimal values
>>7998946
>but what is (definite) integral of xf(x) ?
The area under g(x) where g(x) = x*f(x)
>>7999459
If you know and are comfortable with what a function is, none of this is a problem.
>>7998946
For a continuous random variable [math]Y[/math], where [math]f(y)[/math] is the probability density function, [math]\int^{\infty}_{-\infty} yf(y)dy[/math] is the expected value of [math]f(y)[/math]. Note that [math]f(y)>0\forall y[/math] and the density function must integrate to 1.
>>7999938
It doesn't seem very practical in 2 d calc but in calculus with several variable it's absolutely critical.
Moreover, it does capture some of the intuition behind the definition of the integral.
Your teachers aren't just nagging you when they say you need to be in the habit of writing a dx. It is important.
>>8000083
>Your teachers aren't just nagging you when they say you need to be in the habit of writing a dx. It is important.
You say this like not being anal about a dx means that I'm in high school calculus. It's not uncommon to see the notation [math]\int f[/math] to denote integrating a function [math]f:[a,b] \to \mathbb R[/math], and in fact this is what the book we used for my real analysis classes in undergrad did. It's actually nicer any time we can put distance between a function and a function applied at some indeterminate point. If everyone understands what is being said, why should we waste chalk tacking on an annoying dx at the end of every integral?
>but in calculus with several variable it's absolutely critical.
That is something entirely different from what we are discussing, though. Even then, it's more useful bookkeeping rather than a fundamental feature.
[eqn]
\begin{aligned}
\int x f(x) \; dx &\cdots \quad u = x \;\; du = dx \;\; v = F(x) \;\; dv = f(x) \\
\int x f(x) \; dx &= x F(x) - \int F(x) \; dx
\end{aligned}
[/eqn]
>>7999418
This guy gave the right answer.
Everyone else is circle jerking about what could they desperately google when their highschool teachers weren't looking so that they can post it here and pretend they arent underage b8.
Bravo /sci/.
>>8000096
I'm happy to see /sci/ is still "who knows more math: the board"
You're a HUGE loser, bro, you need better hobbies.
>>8000105
He asked for an interpretation though
>>8000106
His argument was based on talking down to me. I gave an example of the dx being unnecessary, because there is literally no reason to make a huge deal about including it except to try to be smug. Instead of any conversation about it, I'm told that I'm a loser and I'm the one trying to seem smarter than everyone. Go fuck yourself.
>>7999430
Jesus christ, I fucking hate non-Americans. All they learn is how to shitpost and make assumptions without justification, nothing related to the actual empirical process of evidence. Fuck non-America and fuck your retarded education system.
>>7999938
>millennials
>>8000128
An integral is always taken w.r.t a measure.
>>8000096
>it's not uncommon...
yes it is
what fucking university do you go to where it's not relevant?
sure, if you're in one dimension and everyone knows that you might be able to make the case that the dx is unnecessary, but in most courses and all applications, it is necessary.
in fact, anon, even if you're in one dimension the 'dx' remains completely relevant, see the riemann–stieltjes integral.
>>8000175
OH AND LET'S NOT FORGET VARIABLE SUBSTITUTION
The area under the curve of x operating on f(x)
>>8000175
>sure, if you're in one dimension and everyone knows that you might be able to make the case that the dx is unnecessary
Which is literally the case we are discussing. And considering there is some difference between iterated integrals and integrating a function of multiple variables, it's not like you absolutely need it in multiple dimensions. It is certainly helpful when performing iterated integrals, though.
>see the riemann–stieltjes integral.
So if I'm doing something different, then I have to do things differently?
>>8000177
It's a helpful mnemonic for how to change variables, but it's not like I'm unable to change variables because I didn't write dx. You can feel free to use the dx, but there's no reason to have a thread derail this hard because somebody didn't use it.
>>8000128
>talking down to me
It's an ANONYMOUS image board. You can't take being mistaken for a high schooler personally, especially when you're fucking arguing (to entertain yourself??) that you don't need to write a dx!
>>8000137
you triggered me XD
>>8000097 <------ this
unless x is a constant value, then its just
x times the integral of f(x)
>>8000175
Define: [math] f \in {\Omega ^1}M [/math] s.t. [math] f = f\left( x \right)\operatorname{dx} [/math]
Then it makes perfect sense to write [math] \int f [/math].
>>8000299
lol write in english dumb alien
>>7998946
replace the f(x) with a y and you get the integral of x*y dx dy ... take multivariable calculus and learn to do a double integral
>>8000318
go to bed, Billy
>>7999412
It's the width though. You can't sum up the rectangles without dx because there fucking wouldn't be any.
>>8000097
>he assumes f has a primitive
top laugh
>>8000318
don't you have trig homework due tomorrow high schooler
>>7999459
What if you want to integrate everything at once?
>>7998946
The area under the graph xf(x).
Also your image is missing a dx, otherwise it's meaningless.
>>8000299
Yeah, but that's just using a different symbol to express f(x) dx.
You're still writing the dx, you're not omitting it.
>>7998946
what the fuck are you even integrating ?. unless there's a differential inside f(x) your pic is meaningless .
>>8000097
>∫xf(x)dx
but that wasnt OPs integral
its
∫xf(x)
>>8000134
>proud amerilard
top fucking kek
>>8001300
Yeah, but what is it?
How would I know if he's just writing ∫xf(x)?
>>8001303
x
> autism confirmed
>>8001312
And how would I know?
>>7999459
>>7999416
>>7999658
>>8000900
>>7999430
>>7999397
>these many high schoolers and engineers who think they know math because their last "math" class was a B in freshman diffeq
http://www.math.ucla.edu/~tao/preprints/forms.pdf
Just an example of notation. I don't claim to know how to suck cocks for a living, so please don't go claiming you know how real mathematics is done.
>>8002952
>how real mathematics is done
>completely ignores the fact a lebesgue integral, the more common integral, is always taken w.r.t a measure.
>>8002952
>Just an example of notation
Yes, the most commonly used notation, which is generally the one that is being used, so you would assume it to be used in this situation as well.
Now go fuck off.
>>8003370
Not even remotely true.
