Solution:
$\text { Required area }=\int_{-1}^{1} y d x$
$\begin{array}{l}
=\int_{-1}^{1} x|x| d x \\
=\int_{-1}^{0} x^{2} d x+\int_{0}^{1} x^{2} d x
\end{array}$
$\begin{array}{l}
=\left[\frac{x^{3}}{3}\right]_{-1}^{0}+\left[\frac{x^{3}}{3}\right]_{0}^{1} \\
=-\left(-\frac{1}{3}\right)+\frac{1}{3} \\
=\frac{2}{3} \text { sq.units }
\end{array}$
As a result, the correct answer is C.