Suppose that $f$ is the function shown here, and that $f$ is a piecewise function whose parts are ether portions of lines or portions of circles, as pictured.

In addition, let $A$ be the function defined by the rule $A(x)-{\displaystyle \underset{0}{\overset{x}{}}}f(t)dt.$

$($a$)$ What does the Fundamental Theorem of Calculus tell us about the relationship between A and $f?$

$($b$)$ Compute exact values of:

$A(1)=$

$A(3)-$

$($c$)$ Sketch a precise graph of $y-A(x)$ on the blank axes $($above right$)$ that accurately reflects where $A$ is increasing and decreasing, where $A$ is concave up and concave down, and the exact values of $A$ at $x-0,1,$dots,$7.$