How to find integrals using limits? How to find integrals using limits?
The question arise when I see that to find the derivative of a function $f(x)$ we need to find: $$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ and it works fine for finding derivatives of every function you can give. But is there a similar approach to find integrals using limits?
Thanks!

Yes.
The simplest way to write
$\int_a^b f(x)\, dx$
as a limit
is to divide the range of integration
into $n$ equal intervals,
estimate the integral over each interval
as the value of $f$
at either the start, end, or middle
of the interval,
and sum those.
Mathematically,
$$\int_a^b f(x)\, dx
= \lim_{n \to \infty} \frac1{n} \sum_{k=0}^{n-1} f(a+(k+c)\frac{b-a}{n})
$$
where $c = 0, 1/2, 1$
for the function to be evaluated at,
respectively,
the start, middle, or end
of the subinterval.
For a reasonable class of functions,
this limit converges to
the integral.
Generalizations of this
have provided many careers
in mathematics.
