Let $g(x)={\displaystyle \underset{0}{\overset{x}{}}}f(t)dt,$ where $f$ is the function whose graph is shown.

$($a$)$ Evaluate $g(x)$ for $x=0,1,2,3,4,5,$ and $6.$

$($b$)$ Estimate $g(7).$

$($c$)$ Where does $g$ have a maximum value? Where does it have a minimum value?

$($d$)$ Sketch a rough graph of $g.$