I am learning integration right now and I was wondering if there is something like the definition of a derivative but for an integral. So, to find a derivative we can do the following:
$$f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$$
Is there something like that for an integral?
Thanks!
You can write it slightly differently, but this is often taken as the defintion of the definite integral: $$ \int_{a}^{b} f(x) \; dx = \lim_{n\to \infty} \sum_{i=1}^{n}f(x_i)\Delta x $$ Here $\Delta x = \frac{b-a}{n}$ and $x_i = a + i\Delta x$.
To understand this definition better, you could take a look at this Wikipedia article.
There is a relation between the indefinite integral and the definite integral. The Fundamental Theorem of Calculus says that (assuming that $f$ is integrable) $$ \int_a^b f(t)\; dt = F(b) - F(a) $$ where $F$ is any antiderivative (i.e. $ F'(x) = f(x)$). Now given constant $a$ that would mean that $$ F(x) = \int_a^x f(t)\; dt + F(a). $$