Q. If $f(x) = \left\{ \begin{array}{r} |x|, \\ |x - 2|, \end{array}\ \begin{matrix} - 1 \leq x \leq 1 \\ 1 < x \leq 3' \end{matrix} \right.\$ then $\int_{- 1}^{3}{f(x)dx}$ is equal to
A) 0
B) 1
C) 2
D) 4
Explanation
(c)
$\int_{- 1}^{1}{|x|dx + \int_{1}^{3}{|x - 2|dx}}$
$= 2\int_{0}^{1}{x\ dx +}\int_{1}^{2}{(2 - x)dx +}\int_{2}^{3}{(x - 2)dx}$
$= 2 \times \frac{1}{2} + \left\lbrack 2x - \frac{x^{2}}{2} \right\rbrack_{1}^{2} + \left\lbrack \frac{x^{2}}{2} - 2x \right\rbrack_{2}^{3}$
$= 1 + \frac{1}{2} + \frac{1}{2} = 2$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.