Problem $3.{(}^{****})$ Explain the mathematical meaning and purpose of each of the components of a Riemann sum and integral, using complete sentences.

$\underset{n}{lim}{\displaystyle \underset{i=1}{\overset{n}{}}}f(x_i)x={\displaystyle \underset{a}{\overset{b}{}}}f(x)dx$

$($a$)$ The mathematical meaning and purpose of ${\displaystyle \underset{?}{\overset{?}{}}}$ is $...$

$($b$)$ The mathematical meaning and purpose of $x$ is $...$

$($c$)$ The mathematical meaning and purpose of $\underset{n}{lim}$ is $...$

$($d$)$ The mathematical meaning and purpose of ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$ is$...$

$($e$)$ The mathematical meaning and purpose of $dx$ is$...$

$($f$)$ The mathematical meaning and purpose of $+C$ is$...$