$1.(8$ points$)$ A beam carries a distributed load given by $\backslash ($ w$($x$)=2024$ e$^\{-0\mathrm{.}15$ x$\}\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash ),$ as shown. For each part below, write a single labeled expression, label $\backslash (=\backslash )$ formula, that represents the desired quantity. Units not required. Do not compute any integrals.

$($a$)$ What is the total force on the beam?

$($b$)$ What is the moment of a typical slice about the pivot $\backslash ($ C $\backslash )$ at the center of the beam?

$($c$)$ What is the moment of a typical slice about the pivot $\backslash ($ O $\backslash )$ at the origin?

$($d$)$ What is the moment for this beam about a pivot $\backslash ($ A $\backslash )$ at an unknown location $\backslash ($ x$=$a $\backslash )$ in $\backslash ([0,5]\backslash )?$ Provide an exact symbolic answer, which will involve $\backslash ($ a $\backslash ).$