Solution:

$\begin{array}{l}
\text { Required area }=\int_{-2}^{1} y d x \\
=\int_{-2}^{1} x^{3} d x \\
=\left[\frac{x^{4}}{4}\right]_{-2}^{1}
\end{array}$
$\begin{array}{l}
=\left[\frac{1}{4}-\frac{(-2)^{4}}{4}\right] \\
=\left(\frac{1}{4}-4\right)=-\frac{15}{4} \text { units }
\end{array}$
As a result, the correct answer is $\mathrm{B}$.