Meaning of $dx$ in integration and differentiation In differentiation, we denote $dx$ as in infinitesimal change in $x$ and $dy$ the corresponding change.But in integration, we divide $x$ into infinitely many equal intervals and name them $dx$. Then $f(x)dx$ represents the area of an infinitely thin rectangle. From differentiation we know, $dy = f(x)dx$.Why do we substitute then $dy$ in place of $f(x)dx$? I mean $dy$ is change in differentiation. But here in integration we take $dy$ to be the area of each infinitesimal rectangle. Why do we mix them up? Please someone explain with clarity.
$dx$ and $dy$ are not infinitesimal changes, this is an old, obsolete legend. Here is the truth: https://en.wikipedia.org/wiki/Differential_of_a_function.
The answer is &quot;the fundamental theorem of calculus&quot;.  The fundamental theorem of calculus shows the deep connection between slopes and areas.  Because of the fundamental theorem of calculus, $dy$ is both the $y$-component of the slope of $f(x)$ and the area of the individual boxes $f'(x)\,dx$.  That's the fundamental theorem of calculus.
However, I think viewing calculus as fundamentally about slopes and areas is a wrong way of going about it.  I think a simpler way to think about it is this:

Differentiation is about infinitesimal changes
Integration is about adding together infinitesimal changes

Here, you can easily see why integration works: you are adding up changes to get the total change (and the total change will be off from the `true value' by a constant, $+C$).
That is, if I differentiate $x^2$, I will get $2x\,dx$.  This means that the value of the change in $x^2$ depends on both (a) the current value of $x$, and (b) the amount of (infinitesimal) change in $x$.  If I add up all of those changes (I integrate it), I will get the total change.  In other words,
$$\text{total change in }x^2 = \int 2x\,dx$$
If I make a graph of this, it will be the same as the original graph, but perhaps off by a constant - which is precisely what integration does!  So integration gives us a mechanism to add together an infinite number of infinitely small values.
So, how does this relate to areas?
The area under a curve is simply adding together all of the infinitely small rectangles under it.  Each rectangle is $y$ high and $dx$ wide, or $y\,dx$.  Therefore, the sum of all the individual rectangles is $\int y\,dx$.
Now, we know enough to find the relationship between slopes and areas.  To find the slope of an equation, we take the differential and divide by $dx$.  To find the area under the curve of an equation for $y$, we multiply by $y$ and integrate.  Since integration and differentiation are opposites of each other, the process of finding areas is the reverse of the process of finding slopes.