Question

Find the area of the shaded region f(x)= x^3+x^2-6x and g(x)=6x. The bounds are [-4,0].

Answer 1

If you look at the graph below, the "ceiling" function, is f(x), on that interval of [-4,0], whilst the "floor" function, is g(x)

so, taking that into account, then $\bf \begin{cases} f(x)=x^3+x^2-6x\\\\ g(x)=6x \end{cases}\qquad \displaystyle \int\limits_{-4}^{0}\ \begin{array}{llll} [ceiling]&-&[floor]\\ f(x)&&g(x) \end{array} \\\\\\ thus \\\\\\ \displaystyle \int\limits_{-4}^{0}\ [[x^3+x^2-6x]-[6x]]dx\implies \displaystyle \int\limits_{-4}^{0}\ x^3+x^2-12x \\\\\\ \left. \cfrac{x^4}{4}+\cfrac{x^3}{3}-6x^2 \right]_{0}^{-4}$