So this seemed so very cool that I wanted to learn more. There were a few problems, however. The direction of Nishimura is traveling opposite to what this video shows. I eventually found this video is from Comet Encke back in 2007 as observed by STEREO-A. https://en.wikipedia.org/wiki/File:Encke_tail_rip_off.ogv
Well I can't be that mad even though I was tricked because this is a really cool video and I learned some new things. Nishimura did go through a tail disconnection event. But buyer beware, eh?
For number 4 (or maybe a new number), if a(x) and b(x) are constant, the derivatives d/dx a(x) and d/dx b(x) are zero, so all that is left is that d/dx becomes a partial derivative and can move inside the integral.