What Does B Width Mean Integrals; Exactly What Does The "dx" Mean?

Integrals; exactly what does the "dx" mean? - what does b width mean

East:
a) the width of the BA
b) the width of a single sub-rectangle (one of the infinitely small rectangles)

The question is resaons have participated to a deeper understanding of the integrals in Arithmatic. The question was, when substution is ...

∫ sin (2x) dx

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

Now I can replace the DX "in the original equation, blah, blah.

What exactly is DX? Midlessly do not just want to Arithmatic.

Only the hope of an "Aha!" Time here. :-)
Thanks