Dork Scratchings

I've been wondering: is it possible to visualize the rules of calculus? 1. Integration by parts To simplify things, in the above visualization $v(x_1) = 0$, so that $\int^{x'}_{x_1}\frac{dv}{dx}dx = v(x')$.  This means that $v(x_2)$ is simply the area of the cross-section facing us on the LHS, and the volume  on the LHS is $u(x_2)v(x_2)$. The RHS shows the same volume split into two parts.  The rectangle embedded in the first is $u\frac{dv}{dx}$, and so its volume is $\int^{x_2}_{x_1}u\frac{dv}{dx}dx$.  The cross section of the 2nd part is $v(x)$, so its volume is $\int^{u_2}_{u_1}vdu$, which becomes $\int^{x_2}_{x_1}v\frac{du}{dx}dx$ when rewritten as an integral over $x$. So the picture is visual proof that the equation in pink holds, provided that $v(x_1) = 0$.  To prove it in general we just need to check that when we replace $v$ with $v+v_1$ in the equation, both sides change by the same amount. 2. The chain rule This picture demonstr...