Anyone care to explain to me what dx (or any other variable besides

>>8761035
dx is an infinitesimal unit of x. it comes from the definition of a derivative where we increase a function by the smallest possible amount to see how it changes

>>8761036
So would that explain why if you have, say, a function for velocity v(t) in m/sec, you would have integral from a to b of v(t)dt, dt would be in seconds and the seconds would cancel out, leaving you with displacement?

>>8761035
Look up integration by first principle. When you use integration to find the volume of a sphere you are summing the volume of an infinte number of cillinders from 0 to h. You find the area of a cross sectional circle and then times it by dx to find the volume. Do this an infinite number of times and this sum can be rewritten as an integral.

In short, the dx is the 'height' of the cross sections.

>>8761035
When you think back to summing up those rectangles, you used a change in x along the x axis for the base of the triangle when calculating. That distance is called delta x. "dx" is its equivalent. It really isn't much more than a symbol showing you that you're making steps of infinitely small size along the x direction instead of large "delta x" steps. When you do other integrals and see things with dy and dx in your integration, you'll see that it is very important to keep which axis you are integrating along straight, so dx is a big deal. But in early integration, it kind of falls away on its own and everything is convenient so nobody really talks about it.

>>8761070
>times it by
lad...

It's literally just notation.

>>8761081
yeah my bad, old habits die hard

File: 14899390023321153701547.jpg (4MB, 5312x2988px) Image search: [Google]

4MB, 5312x2988px

>>8761070
for example, surface area of a cone

>>8761035
as anon already said, it's just notation. I may be wrong on this one, but when this was invented (Leibniz?), they actually treated the dx's as objects on their own, but it was non-rigorous, it just "somehow" worked. the "somehow" is now well understood - manipulation with differentials is actually just chain rule and linearity of the derivative operator (which are basic theorems in calculus) in a fancy disguise. this notation is extremely convenient so it is still used but really there is no such thing as "multiplying by the dx".

>>8761125
also.. if this keeps bugging you after you take multivariable calculus, dive yourself into differential geometry. the dx's are given a precise meaning: they are called differential forms and very roughly they are something like a infinitesimal k-dimensional volume - which can be integrated over a k-dimensional space.

>>8761145
I think that's a long way to go for me, since I'm taking the equivalent of a first year calculus college course so there's not as much as I would like. Nevertheless it's something that definitely sounds interesting to me despite not knowing what it is. Don't know why but I just really like calculus

>>8761035
As addition to other posts upside, remember that d in [math]\int f(x)\text{d} (x)[/math] is an operator. Not a variable.

>dx is an infinitesimal

Fucking REEEEEEEEEEE

Look up differential forms, and look up total differential.

>>8761035
f(x) = length of the rectangle at some x
dx = width of the rectangle
area of a rectangle: length * width

let dx be an infinitesimally small value
multiply it by the height at some x to calculate the area of the rectangle at that x
do it for all x from a to b.
sum all the areas up

it actually IS that simple.

>>8761266
No it isn't. dx is a measure.

>>8761285
good point D:

>>8761267
That's totally disingenuous. We've been writing dx in integrals long before the advent of modern differential geometry. That theory was mostly put together just to fit in with notation we already had.

>it's just notation

Then why are you allowed to differentiate a function and replace the x with it behind the d when integrating by parts?

>>8761035
You may want to look into what is called the hyperreals.

Everybody in this thread except for >>8761145
is retarded. Basically, under the integral sign it's just notation, but !!!non-rigorously!!!, you might think of it as summing all the rectangles of infinitesimal width. On it's own, you can either treat it as a differential or a differential form. Per partes is directly acquired from the differentiation of a product of functions. Substitution stems from chain rule, as was mentioned previously.

>>8762412
This is all based upon the definition of a primitive function. Equivalence of Riemann and Newton integrals then gives you both the intuition of summing rectangles and also allows you to use the tools you have from differential calculus.

>>8762391
>hyperreal

Non-non-standard analysis BTFO

>>8761108
what ink/pens did you use for these notes? I like the shades