<s>[INST] The area bounded by the curve questions? What's the area bounded by the curve $f(x)=3x^3 - x^2 - 10x$ and $g(x)=-x^2+2x$
I solved for $f(x) = 0$ and get $-\frac{5}{3}$ , $0$ , $2$
and $g(x) = 0$ and get $0 , 2$
Solved $f(x) = g(x)$ I Found their third intersection when $x = -2$
The problem I've is how to solve the area with all these known intersections ? 
 [/INST] 
We separate the integral into two integrals to find the area bounded by the two curves which intersect at $x = -2, x = 0, x = 2$: subtracting the upper curve from the lower curve.
$$\int_{-2}^0 (3x^3 - x^2 - 10 x - (-x^2 + 2x))\,dx + \int_{0}^2 (-x^2 + 2x - (3x^3 - x^2 - 10 x))\,dx$$
Simplifying gives us $$\int_{-2}^0 (3x^3 - 12x)\,dx + \int_{0}^2 (-3x^3 + 12x)\,dx$$
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