What does it mean by the Integration notation? $\int_a^b f(x)  dx = 1 -------------------(*)$
I know that this means:
(1) Integral of $f(x)$ within $a$ and $b$ is $1$.
or, 
(2) The area under the curve represented by the function $f(x)$  between the interval $a$ and $b$ is $1$. 
But, my question is, what is $dx$ doing in the equation $(*)$?
And, how is it effecting the result of the equation?
What is the value of $\int_a^b f(x)$ in this context (i.e. without $dx$)?
 A: I had this question some days and I got my answer when I read this book 
One Variable Calculus with an Introduction to Linear Algebra by Tom. M. Apostol
The thorough answer to this question is given in that book. Specifically, see these parts of the book
1) Section 4.7, Other notations for derivatives
2) Section 5.6, The Leibniz notation for primitives
A: What dx is telling you is that you are integrating with respect to x.  What this means is that it defines how you are finding the area.  
When you said you were finding the area under the curve, you were finding the area between the line f(x) and the x-axis.  That dx is there to note that you're using the x-axis to find area.  
You could switch out the dx for dy and find the area between the line f(x) and the y axis, but that's probably a couple units down the road for you.  
If you left out the dx, it would be incorrect, as you must know what you are deriving with respect to e.g. which axis you're using for area.
That's the calculus for dummies way of putting it.
