$(8$ points$)$ A beam carries a distributed load given by $w(x)=2024e^{-0\mathrm{.}15x}\frac{N}{m},$ as shown. For each part below, write a single labeled expression, label $=$ 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 $C$ at the center of the beam?

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

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