Let $g(x)$ be the function whose graph is shown below.

a$)$ Evaluate ${\displaystyle \underset{0}{\overset{3}{}}}g(x)dx$ and ${\displaystyle \underset{0}{\overset{6}{}}}g(x)dx.$

b$)$ Evaluate ${\displaystyle \underset{0}{\overset{8}{}}}g(x)dx$ and ${\displaystyle \underset{3}{\overset{9}{}}}g(x)dx.$

c$)$ Let $A(y)={\displaystyle \underset{0}{\overset{y}{}}}g(x)dx.$ Explain why $A(y)$ has a local maximum at $y=3.$ Where are the other relative extrema of $A(y)?$