How can I get the function of a curve from a dataset without using a curve fitting tool?
It's not necessary to first fit a curve to the data and then compute the integral. You can directly approximate the integral from the data using numerical integration methods. The most straightforward approximation would be a Riemann sum:
$$S=\sum_ny_n\Delta x\tag{1}$$
where $y_n$ are your data, and $\Delta x$ is the distance between the individual data points on the $x$-axis (assumed constant here, but that's not a necessity).
Of course there are more advanced methods of numerical integration, but $(1)$ is very simple and might be sufficient for your application.