Question

The value of ${\int }_0^2|x^2-1|dx$ is

• A. $0$
• B. $2$
• C. $–\left(\frac{1}{3}\right)$
• D. $-2$

Answer 1

The correct option is B

$2$

Explanation for the correct answer.

Evaluate the integral.

$\begin{array}{rcl}{\int }_0^2|x^2-1|dx& =& {\int }_0^1|x^2-1|dx+{\int }_1^2|x^2-1|dx\\ & =& {\int }_0^1\left(1-x^2\right)dx+{\int }_1^2\left(x^2-1\right)dx\\ & =& {\left[x-\frac{x^3}{3}\right]}_0^1+{\left[\frac{x^3}{3}-x\right]}_1^2\\ & =& \left(1-\frac{1}{3}\right)-\left(0-\frac{0}{3}\right)+\left(\frac{2^3}{3}-2\right)-\left(\frac{1}{3}-1\right)\\ & =& \frac{2}{3}+\frac{2}{3}-\left(\frac{-2}{3}\right)\\ & =& 2\end{array}$

Hence, $o\mathrm{ption}\left(\mathrm{B}\right)$ is the correct answer.