Basic integrations work with integral which defined like :
1 $$\int_0^1f(x)dx = \lim_{n\to\infty}\sum_{k=1}^nf\left(\frac{k}{n}\right)\frac{1}{n}$$
But why we need and consider Riemann integrals?
almost integrations work with equal division like method 1.
I just think this (1) can't make sense for discontinuous function $f$.
But this is just my think.
What I want to know:
(1) Why we consider Riemann Integral instead of 1?
(2) What advantages we get if we take Riemann Integral instead of 1?