$S=\displaystyle\int\limits^1_0 |x^4-4x^2+4-x^2| \, dx\\ =\displaystyle\int\limits^1_0 |x^4-5x^2+4| \, dx\\ f(x)=x^4-5x^2+4=(x^2-4)(x^2-1)=(x-2)(x+2)(x-1)(x+1)\\ BXD:\\\begin{array}{|c|ccccccccccc|} \hline x&-\infty&&-2&&-1&&0&&1&&2&&+\infty\\\hline f(x)&&+&0&-&0&+&|&+&0&-&0&+\\\hline\end{array}\\ \Rightarrow f(x) \ge 0 \, \forall \, x \in [0;1]\\\Rightarrow S=\displaystyle\int\limits^1_0 (x^4-5x^2+4) \, dx\\ =\left(\dfrac{x^5}{5}-5\dfrac{x^3}{3}+4x\right)\Bigg\vert^1_0\\ =\dfrac{38}{15}$
