If you were to try to approximate the value of $f(x+3\Delta x)$, you could use Euler's method to make 3 linear approximations.
$$f(x + \Delta x) \approx f(x) +\frac{df}{dx}(x)*\Delta x $$
$$f(x + 2\Delta x) \approx f(x+\Delta x) +\frac{df}{dx}(x+\Delta x)*\Delta x $$
$$f(x + 3\Delta x) \approx f(x+2\Delta x) +\frac{df}{dx}(x+2\Delta x)*\Delta x $$
From this method, you can get that
$$f(x + n \Delta x) \approx f(x) + \sum_{i=0}^{n-1} \frac{df}{dx}(x+i\Delta x) \Delta x$$
Where given n-linear approximations, the width $\Delta x$ is calculated as $\frac{b-a}{n}$.
If you take an infinite limit, it's equivilant to the Riemann integral limit, and you get
$$f(x) + \int \frac{df}{dx} dx$$
$$\int f'(x) dx + C$$
Is there a name for the limit definition of an integral, where the construction appears like Euler's method for approximation, until you evaluate it with a limit as you would do for a Riemann approximation that then evaluated with a limit gives an exact answer?
Thanks!