Comparing Regularization Terms in Regression Models When learning about Regularization in Regression, I am used to seeing this in the following form (https://en.wikipedia.org/wiki/Regularization_(mathematics)):

When watching this video (https://www.youtube.com/watch?v=0nYiGXooG4I), I see a similar equation as provided above:

The thing which I don't understand is that when comparing the second equation to the first equation - why is there an integral in the second equation and why is there a second derivative of the function?
Thanks!

This video talks about fitting smoothing splines, so they want the splines in $f$ to be smooth. Since $f^{\prime\prime}$ is measuring the curvature of $f$, $\int (f^{\prime\prime}(x))^2dx$ is (proportional to) the average squared curvature.
So, this optimization penalizes large average curvature; apparently, smooth here means to be rather flat.
