Understanding the Relationship Between Integration and Anti-differentiation

[Muthumanimaran]

While attempting proof for Integration is anti-differentiation from the book Mathematical methods for physics, Riley, Hobson
How the R.H.S of the equation (after rearranging and dividing by Λx) becomes f(x)Λx?

 

[Svein]

How the R.H.S of the equation (after rearranging and dividing by Λx) becomes f(x)Λx?

Well, the (Riemann) integral from a to b is defined as the upper limit of $\sum_{\bigcup \Delta x = [a, b]}f(x)\cdot\Delta x$. So, if you put a=x and b=x+Δx, you get the first element in the sum. Which means that $\int_{x}^{x+\Delta x}f(x)dx$ is the upper limit of $f(x)\cdot \Delta x$ (as Δx→0).