6.1 Areas Between Curves/10

${\begin{aligned}&\color {red}\mathbf {y=1+{\sqrt {x}}} &\color {royalblue}\mathbf {y={\frac {3+x}{3}}} \\\\\end{aligned}}$

${\begin{aligned}&1+{\sqrt {x}}={\frac {3+x}{3}}\\&1+{\sqrt {x}}-{\frac {3+x}{3}}=0\\&{\frac {3+3{\sqrt {x}}}{3}}-{\frac {3+x}{3}}=0\\&3+3{\sqrt {x}}-3+x=0\\&3{\sqrt {x}}+x=0\\&3{\sqrt {x}}=-x\\&9x=x^{2}\\&9x-x^{2}=0\\&x(9-x)=0\\&x=0,9\end{aligned}}$

$\int _{0}^{9}\left(1+{\sqrt {x}}-{\frac {3+x}{3}}\right)dx=\int _{-3}^{-2}\left((x^{2})-(8-x^{2})\right)dx+\int _{-2}^{2}\left((8-x^{2})-(x^{2})\right)dx+\int _{2}^{3}\left((x^{2})-(8-x^{2})\right)dx={\frac {14}{3}}+{\frac {64}{3}}+{\frac {14}{3}}={\frac {92}{3}}$