$(15$ pts$)$ A long, thin rod lies along the x$-$axis. One end of the rod is located at x $=1\mathrm{.}00$ m$,$ and the other end of the rod is located at x $=3\mathrm{.}00$ m$.$ The linear mass density of the rod is given by $($x$)=$ ax$2+$ b $+$ ce$$dx$,$ where a $=0\mathrm{.}350$ kg$/$m$3,$ b $=0\mathrm{.}625$ kg$/$m$,$ c $=0\mathrm{.}150$ kg$,$ and d $=0\mathrm{.}0500$