integrals; exactly what does the "dx" mean?

Question

Is it:
a) the width of b-a
b) the width of a single sub-rectangle (one of those infinite tiny rectangles)

The resaon I ask is to get a better deeper understanding of the arithmatic involved in integrals. The question occured when doing substution...

∫sin(2x) dx

substitute:
u=2x
du/dx = 2
dx = 1/2 du

now I replace the "dx" in the original equation, yada yada.

So what exactly is dx? I don't want to just midlessly do arithmatic.

Just hoping for an 'aha!' moment here. :-)
Thanks

Answer 1

Haha, i was searching about it somedays ago as i just started studying it and i found one that explains it really easily :)

Check the source.

Answer 2

It is an infinitesimal. The integral of a function is the area of the curve to the x-axis. It is the infinite sum of infinitely many, infinitely small strips. The width of a strip is dx. The height of the strip is f(x).
In the case of derivatives, dy/dx is an infinity small triangle on the curve at (x, y) where y = f(x). The hypotenuse is tangent to the curve and in fact, dy/dx is the tangent of that little triangle and therefore the tangent to the curve; meaning that if you extend the hypotenuse to the x-axis, and draw another line straight down to the x-axis, you will have a triangle.
Then tan(angle) = y/x where x here means, not the x-ordinate, but the length of the base of your triangle. The fundamental theorem of calculus is the amazing coincidence that the derivative of the integral is the function. Or to put it another way, that the area under a curve and the slope of a tangent line to a curve are inverse.
Another infinitesimal is arc length (ds)^2 = (dx)^2 + (dy)^2.
Well, I use the term infinitesimal, but in fact it is spoken of as such as little as possible in calculus. They always speak of everything in terms of limits. But the term infinitesimal is a nice alternative to using an entire sentence involving limits in every instance in speaking about them.

Answer 3

dx means that you "integrate with respect to x"
I = ∫ sin (2x) dx
let u = 2x
du = 2 dx
du/2 = dx
I = (1/2) ∫ sin u du
Have now to integrate with respect to u:-
I = (1/2) (- cos u) + C
I = (-1/2) cos 2x + C

Answer 4

good question.
dx actually means the difference between two values of x which is a differential distance, i.e. very very small distance.
that means you are integrating a large number of dx to find the area or anything

Hope it helps