Proof that Integration is opposite of differentiation

Question

In Calculus we learnt to find derivative as $$ f'\left(x\right) = \lim_{h \rightarrow 0} \frac{f\left(x+h\right)-f\left(x\right)}{h} $$ and integration as $$ \int_a^bf\left(x\right)dx=\lim_{n \rightarrow \infty} \sum_{r=1}^nf\left(t_r\right)\phi_r $$ But these definitions in no manner seem to suggest that they might be opposite of each other but they are.
can we just prove that these to processes are just opposite of each other
ie. if we put $f\left(x\right)$ as $f'\left(x\right)$ in second expression will we get $f\left(x\right)$

Answer 1

Indeed they are.

For a sequence that is equally spaced by $h$, namely $u =(u_0,u_1,...)$ Its derivative is given by the application of the operator $D$: $$D = \frac{1}{h}\left[\begin{matrix} 1 & & & \\ -1 & 1 & &\\ & -1 & 1 & \\ & & \ddots& \ddots \end{matrix}\right]$$ This implicitly sets $u_0=0$ (as usual for a derivative operator that must come with a BC), therefore its is applied to $\tilde{u}=(u_2,u_2,...)$.

On the contrary the integral operator is defined as: $$I = h\left[\begin{matrix} 1 & & & \\ 1 & 1 & &\\ 1 & 1 & 1 & \\ \vdots & \ddots & \ddots& \ddots \end{matrix}\right]$$ You can see that one is the inverse of the other, i.e. $\mathbb{I} = ID=DI$, where $\mathbb{I}$ is the identity operator.