Area in the Unit Square

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Question

Find the area of the region enclosed between the axes, the line $y=c−x$, and the unit square (i.e. a square with corners $(0, 0), (0,1), (1,1), (1,0)$)


Hints

Consider the different cases. For example, below is shown the case where $ 1 < c < 2 $:

Area under curve; case where 1 < c < 2


Answer

There are 4 cases. These are summarised below: $$ A = \begin{cases} 0 &\text{if } c \leq 0 \\ \frac{1}{2} c^2 &\text{if } 0 < c \leq 1 \\ 1 - \frac{1}{2} (2-c)^2 &\text{if } 1 < c \leq 2 \\ 1 &\text{if } c > 2 \end{cases} $$