y = 12 − x 2 {\displaystyle \color {red}{y=12-x^{2}}}
y = x 2 − 6 {\displaystyle \color {blue}{y=x^{2}-6}}
∫ − 3 3 | ( 12 − x 2 ) − ( x 2 − 6 ) | d x {\displaystyle \int _{-3}^{3}\left|(12-x^{2})-(x^{2}-6)\right|dx}
12 − x 2 = x 2 − 6 18 = 2 x 2 9 = x 2 ± 3 = x {\displaystyle {\begin{aligned}12-x^{2}&=x^{2}-6\\18&=2x^{2}\\9&=x^{2}\\\pm 3&=x\end{aligned}}}
∫ − 3 3 | ( 12 − x 2 ) − ( x 2 − 6 ) | d x = [ ( 12 x − 1 3 x 3 ) − ( 1 3 x 3 − 6 x ) ] | − 3 3 = [ ( 12 ( 3 ) − 1 3 ( 3 ) 3 ) − ( 1 3 ( 3 ) 3 − 6 ( 3 ) ) ] − [ ( 12 ( − 3 ) − 1 3 ( − 3 ) 3 ) − ( 1 3 ( − 3 ) 3 − 6 ( − 3 ) ) ] = 36 − ( − 36 ) = 72 {\displaystyle {\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}}}
![{\displaystyle {\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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/07815796d9706324e8cf47b43addaf93fd5af522)
