TITLE: Find area enclosed?
QUESTION [0 upvotes]: Surface $S_1$ is formed by $|x-1|≤ 1$ and $|y-2|≤ 3$. Surface $S_2$ is formed by $x ≥3$, $y ≤1$ and $x+2y ≤3$. S is union of $S_1$ and $ S_2$. 
Find the area of surface $S_1$, $S_2$ and $S$ ?

REPLY [1 votes]: $S_1$ is a rectangle centered at $(1,2)$ with sides of length $2$ and $6$ respectively. Therefore $A = 12$
$S_2$ is not a closed polygon. Therefore its area is infinite. Check that $x \geq 3$ and $y \leq 1$ makes any value from $x\in (3,\infty)$ and $y \in (-\infty,1)$ part of it. Then $y \leq \dfrac{3}{2} - \dfrac{x}{2}$ excludes only some upper values but all below the line $y = \dfrac{3}{2} - \dfrac{x}{2}$ will still be valid.
The union of both areas will be an infinite area as well.
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