6.1 Areas Between Curves/13

$\color {red}{y=12-x^{2}}$

$\color {blue}{y=x^{2}-6}$

$\int _{-3}^{3}\left|(12-x^{2})-(x^{2}-6)\right|dx$

${\begin{aligned}12-x^{2}&=x^{2}-6\\18&=2x^{2}\\9&=x^{2}\\\pm 3&=x\end{aligned}}$

${\begin{aligned}\int _{-3}^{3}\left|(12-x^{2})-(x^{2}-6)\right|dx&=[(12x-{\frac {1}{3}}x^{3})-({\frac {1}{3}}x^{3}-6x)]{\Bigg |}_{-3}^{3}\\&=[(12(3)-{\frac {1}{3}}(3)^{3})-({\frac {1}{3}}(3)^{3}-6(3))]-[(12(-3)-{\frac {1}{3}}(-3)^{3})-({\frac {1}{3}}(-3)^{3}-6(-3))]\\&=36-(-36)\\&=72\end{aligned}}$