A) 50
B) 100
C) 200
D) none of these
Correct Answer: A
Solution :
\[I=\int\limits_{-100}^{100}{f\,(x)\,dx=200\int\limits_{0}^{1}{f\,(x)\,dx}}\] |
(\[\because \]1 is the period of f(x)) |
\[f\,(x)=\left\{ \begin{matrix} \{x\} & 0\le x<\frac{1}{2} \\ \{-x\} & \frac{1}{2}\le x\le 1 \\ \end{matrix} \right.=\left\{ \begin{matrix} x\,\,\,\,; & 0\le x<\frac{1}{2} \\ 1-x & \frac{1}{2}\le x\le 1 \\ \end{matrix} \right.\] |
\[I=200\,\,\left( \int\limits_{0}^{1/2}{x\,dx+\int\limits_{1/2}^{1}{(1-x)\,dx}} \right)=200\] |
\[\left( \frac{1}{8}+\left( 1-\frac{1}{2} \right)-\frac{1}{2}\left( 1-\frac{1}{4} \right) \right)=200\,\,\left( \frac{1}{8}+\frac{1}{8} \right)\] |
= 50 |
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