6.1 Areas Between Curves/13

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \color{red}{y=12-x^2}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \color{blue}{y=x^2-6}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{-3}^{3} \left|(12-x^2) - (x^2-6) \right| dx }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} 12-x^2 &= x^2-6 \\ 18 &= 2x^2 \\ 9 &=x^2 \\ \pm3 &= x \end{align} }

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