6.1 Areas Between Curves/10

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

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

Failed to parse (syntax error): {\displaystyle \int_{0}^{9} \left(1+\sqrt{x} - \frac{3+x}{3}\right)dx = x + \frac{2x^\frac{3}{2}}3\Bigg|_{0}^{9} - \frac{1}3\int_{0}^{9}3+xdx = x + \frac{2x^\frac{3}{2}}3\Bigg|_{0}^{9} - \frac{{1}}3(\frac{{x^2}}3+3x)\Bigg|_{0}^{9} = 9 + \frac{{2(27)}}3 - \frac{{9^2}}2(3) - 9 = \frac{54}}3 - \frac{81}}6 = \frac{108}}6 - \frac{81}}6 = \frac{27}}6 = \frac{9}}2 }