TITLE: Is there a geometric intuition for integration by parts?
QUESTION [35 upvotes]: Is there a geometric intuition for integration by parts?
$$\int f(x)g'(x)\,dx = f(x)g(x) - \int g(x)f'(x)\,dx$$
This can, of course, be shown algebraically by product rule, but still where is geometric intuition? I have seen geometry of IBP using parametric equations but I don't get it.
Newest edit: few similar questions has been asked before, but they use parametric equations to show geometry behind IBP. I am interested if there is geometric intuition which uses functions in Cartesian plane or some other, maybe more natural, explanation.

REPLY [8 votes]: In light of @Adayah's observations, I'll offer a different geometric intuition for $fdg=d(fg)-gdf$, which integrates to the desired result. Consider the special case $f,\,g,\,df,\,dg>0$, so we can draw an $f\times g$ rectangle inside an $(f+df)\times (g+dg)$ rectangle. Apart from a negligible $df\times dg$ corner piece, the trimming $d(fg)$ outside the slightly smaller rectangle is two rectangles of areas $fdg,\,gdf$.