\[{\text{Найдите площадь фигуры}}{\text{, ограниченной кривой }}{\left( {y - 2{x^2}} \right)^2} + {x^2} = 2.\]
\[\begin{array}{l}
{\left( {y - 2{x^2}} \right)^2} + {x^2} = 2 \Leftrightarrow y = 2{x^2} \pm \sqrt {2 - {x^2}} \hfill \\
{y_1} = 2{x^2} + \sqrt {2 - {x^2}} \hfill \\
{y_1} = 2{x^2} - \sqrt {2 - {x^2}} \hfill \\
S = \int\limits_{ - \sqrt 2 }^{\sqrt 2 } {\left( {{y_1} - {y_2}} \right)dx} = 2\int\limits_{ - \sqrt 2 }^{\sqrt 2 } {\sqrt {2 - {x^2}} dx} \hfill \\
\end{array}\]
\[2\pi \]