How can I do almost any integration with the Leibniz and Euler's intuitive concept that integration is the sum of infinite number of "Differentials"?

Question

Is it correct that integration of $f(x)dx$ with limit $'a'$ to $'b'$ $$=f(a)dx+f(a+dx)dx+f(a+2dx)dx+f(a+3dx)dx+...+f(b-dx)dx?$$Can we calculate any integration with only this view (not use any integration rule as 'Anti differentiation')?Euler intuitively stated 'Integration' as 'Anti-Differentials'(not anti derivative).

Answer 1

If $f:\>[a,b]\to{\mathbb R}$, or $\to{\mathbb C}$, is continuous then one can prove that $$\int_a^b f(x)\>dx=\lim_{N\to\infty}{b-a\over N}\>\sum_{k=1}^N f\left(a+{k(b-a)\over N}\right)\ .\tag{1}$$ But it is only in special cases that an integral can be computed in this way. Examples are linear functions $f(x):=px+q$, where you have to sum an arithmetic series, or $f(x):=x^2+px+q$, if you know the formula for the sum of all squares from $1$ to $N^2$. Other examples can be related to geometric series, e.g. $f(x):=e^{\lambda x}$, whereby $\lambda$ may also be complex. In this way you can, e.g., compute the integral $\int_0^{\pi/2}\sin x\>dx=1$.

It is the fantastic power of the FTC that it allows the computation of an integral $\int_a^b f(x)\>dx$ by purely "algebraic" formulas whenever you are able to name a primitive $F$ of $f$.