Calculamos los interceptos de las dos curvas para saber los límites de la region sombreada:
y₁(x) = y₂(x)
-x² + 2 = x
x² + x - 2 = 0
(x - 1)(x + 2) = 0
x₁ = 1 x₂ = -2
Luego:
[tex]{\displaystyle A = \int_{-2}^{1}(y_2(x)- y_1(x))dx }[/tex]
[tex]{\displaystyle A = \int_{-2}^{1}(-x^2+2-x)dx }[/tex]
[tex]{\displaystyle A=-\left[\dfrac{x^3}{3}-\dfrac{x^2}{2}-2x\right]_{-2}^{1} }[/tex]
[tex]{\displaystyle A=\left(-\dfrac{1^3}{3}-\dfrac{1^2}{2}+2(1)\right)- \left(-\dfrac{(-2)^3}{3}-\dfrac{(-2)^2}{2}+2(-2)\right) }[/tex]
[tex]{\displaystyle A=\dfrac{7}{6}-\left(-\dfrac{10}{3}\right)}[/tex]
[tex]{\displaystyle A=\dfrac{9}{2} = 4.5}[/tex]
