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Addition and Subtraction of a Matrix

Let f: R →Rfx...

Question

Let f:R→Rf(x){|x−2[x]|,[x]isodd|2x−[x+1]|,[x]iseven}
where [.] denotes greatest integer function, then ∫−22f(x)dx is equal to

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Solution

$f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} | 2 x + 1 | & - 2 \leq x < - 1 | x + 2 | & - 1 \leq x < 0 | 2 x - 1 | & 0 \leq x < 1 | x - 2 | & 1 \leq x < 2 \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭$

So required Area is $= \frac{1}{2} \times 1 \times 2 + 1 + 1 + \frac{1}{2} \times \frac{3}{2} + \frac{1}{2} \times \frac{1}{2} \times 1$

= $\frac{9}{2}$

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