Lời giải

\( I = \int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt[3]{(1 - x)^2} \, dx = -\int_{-\frac{1}{2}}^{\frac{1}{2}} - (1 - x)^{\frac{2}{3}} \, dx \)

   \( = -\frac{3}{5} (1 - x)^{\frac{5}{3}} \Bigg|_{-\frac{1}{2}}^{\frac{1}{2}} = -\frac{3}{5} \cdot \frac{1}{2} \cdot \sqrt[3]{\frac{1}{4}} + \frac{3}{5} \cdot \frac{3}{2} \cdot \sqrt[3]{\frac{9}{4}} \)

   \( = \frac{9}{20} \sqrt[3]{18} - \frac{3}{30} \sqrt[3]{2} \)

\( \Rightarrow \begin{cases} a = -\frac{3}{20} \\ b = \frac{9}{20} \end{cases} \Rightarrow a^2 + b^2 = \frac{9}{40} \)