<s>[INST] Area bounded by the graphs $y = x^2 – 2x – 6$ and $y = 6 – x^2$? Find the area bounded by the graphs of the following functions 
$$y = x^2 – 2x – 6$$ and $$y = 6 – x^2$$
Are the points $-3$ and $2$? Where to go form there?
 [/INST] The points of intersection can be found by solving
$$x^2-2x-6=6-x^2 \Longleftrightarrow x^2-x-6=0 \Longleftrightarrow (x-3)(x+2)=0$$
Thus $x=\color{red}-2$ and $x=\color{red}+3$ are the points of intersection.
In the interval $[-2,3]$, $6-x^2>x^2-2x-6$, therefore the area of the bounded region is
$$\int_{-2}^36-x^2-(x^2-2x-6)\,dx$$

$$=\displaystyle \int_{-2}^3 -2x^2+2x+12 \,dx$$
$$=\displaystyle \left. \left(-\frac{2}{3}x^3+x^2+12x\right)\right|_{-2}^3$$
$$=\boxed{41\frac{2}{3}}$$

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