[tex]x^2=2+x\\
x^2-x-2=0\\
x^2+x-2x-2=0\\
x(x+1)-2(x+1)=0\\
(x-2)(x+1)=0\\
x=2 \vee =-1\\\\
\displaystyle
A=\int \limits_{-1}^22+x-x^2\, dx\\
A=\left[2x+\dfrac{x^2}{2}-\dfrac{x^3}{3}\right]_{-1}^2\\
A=2\cdot2+\dfrac{2^2}{2}-\dfrac{2^3}{3}-\left(2\cdot(-1)+\dfrac{(-1)^2}{2}-\dfrac{(-1)^3}{3}\right)\\
A=4+2-\dfrac{8}{3}-\left(-2+\dfrac{1}{2}+\dfrac{1}{3}\right)\\
A=6-\dfrac{8}{3}+2-\dfrac{1}{2}-\dfrac{1}{3}\right)\\
A=8-\dfrac{9}{3}-\dfrac{1}{2}\\
A=8-3-\dfrac{1}{2}\\
A=5-\dfrac{1}{2}\\
A=4\dfrac{1}{2}
[/tex]
