Answer
(a)
$$A(2)=4.$$
(b)
$$A(6) =16.$$
(c)
$$A(x)=\Bigg\{_{\frac{x^2}{2}-x+4,\quad x>2.}^{-\frac{x^2}{2}+3x,\quad x\leq2;}$$
Work Step by Step
(a) In this case, the region will be a trapezoid bounded by the $y$ axis, the $t$ axis the graph of the function and the vertical line $t=2$. This line will intercept the graph of the function at the point $(2,f(2))=(2,|2-2|+1)=(2,1)$. This trapezoid has the bases of length of $3$ and $1$ and the height of $2$ so the area is
$$A(2)=\frac{3+1}{2}\cdot2=4.$$
(b) In this case we will split the region into the trapezium determined by $0\leq t\leq2$ which has the area $A(2)$ and the trapezium bounded by $t$ axis, the graph of the function and vertical lines $t=2$ and $t=6$. It has the bases of $1$ and $f(6)=|6-2|+1=5$ while its height is equal to $6-2=4$. The required area is
$$A(6) = A(2) + \frac{1+5}{2}\cdot4=4+12=16.$$
(c) For $x\leq2$ the region bounded by the $y$ and $t$ axes, the graph of the function and the vertical line $t=x$ is a trapezoid with the bases of $3$ and $f(x)=|x-2|+1=2-x+1=3-x$ and the heght equal to $x$ so the area is
$$A(x)=\frac{3+3-x}{2}\cdot x=-\frac{x^2}{2}+3x.$$
If $x>2$ then we have to add to the area $A(2)$ the area of the trapezoid bounded by $t$ axis, the graph of the function and the vertical lines $t=2$ and $t=x$. This trapezoid has the bases of $1$ and $f(x)=|x-2|+1=x-2+1=x-1$ and the height of $x-2$ so the required area is
$$A(x)=A(2)+\frac{1+x-1}{2}\cdot(x-2)=\frac{x^2}{2}-x+4.$$
The final answer is
$$A(x)=\Bigg\{_{\frac{x^2}{2}-x+4,\quad x>2.}^{-\frac{x^2}{2}+3x,\quad x\leq2;}$$
