6.1 Areas Between Curves/16

16)

${\begin{aligned}&\color {red}\mathbf {y=x^{3}-x} &\color {royalblue}\mathbf {y=3x} \\\end{aligned}}$

${\begin{aligned}x^{3}-x&=3x\\x^{3}&=4x\\x^{2}(x)&=4(x)\\x^{2}&=4\\x&={\sqrt {4}}\\x&=\pm 2\\[2ex]\int _{-2}^{2}[(x^{3}-x)-(3x)]dx\\=\int _{-2}^{2}[x^{3}-4x]dx\\[2ex]=\left[{\frac {x^{4}}{4}}-2x^{2}\right]{\Bigg |}_{-2}^{2}\\[2ex]={\frac {(2)^{4}}{4}}-2(2)^{2}-\left({\frac {(-2)^{4}}{4}}-2(-2)^{2}\right)\\[2ex]={\frac {16}{4}}-2(4)-({\frac {16}{4}}-2(4))\\[2ex]=4-8-(4-8)\\[2ex]=-4-(-4)\\[2ex]=-4+4\\[2ex]=0\end{aligned}}$

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