I can almost see how to derive the formula of integration by parts by this extremely helpful picture:
But Im having trouble figuring out how the limits of integration simplify so nicely to how they are, may I please get an explanation on how p,q,s,and r transform into a and b.
I think it is the opposite, i.e. $a$ and $b$ are transformed into $p,q,r,s$. In a parametric form of a curve, one must have the interval for the parameter to vary. So $x\in [a,b]$ is given $$ \left\{ \begin{array}{l} u=f(x),\\ v=g(x) \end{array} \right.\qquad x\in[a,b]. $$ Then $x=a$ gives the starting point of the curve $[f(a),g(a)]$ which is called for simplicity $[p,r]$ and similar for the final point $x=b$.
