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The blue line in the $t,y$ plane shown above has the equation $y=f(t).$ This defines the linear function $f.$

$($a$)$ Consider the function

$A(x)={\displaystyle \underset{0}{\overset{x}{}}}f(t)dt$

Find an algebraic expression for $A(x)$ valid for any $x>4$ :

$A(x)=$

$($b$)$ Find $F(t),$ the antiderivative for the function $f(t)$ that obeys $F(0)=0.$

$F(t)=$