Example 7. Calculate the area of the figure bounded above by the lines $x-y+2=0, y=2-\frac{1}{2} x$, and below by the parabola $y=x^{2}-2 x-$ $-8$.
Solution. Referring to Fig. 3.9. The equation of $AC$ is $y=x+2$, and $CB$ is described by the equation $y=2-\frac{1}{2} x$. Then,

$$
S=\int_{-2}^{0} d x \int_{x^{2}-2 x-8}^{x+2} d y+\int_{0}^{2} d x \int_{x^{2}-2 x-8}^{2-x / 2} d y=42
$$

Answer. $S=42$.