Question Number 199471 by MathematicalUser2357 last updated on 04/Nov/23
$${Find}\:{the}\:{integral} \\ $$$$\int_{−\mathrm{3}} ^{\mathrm{3}} \begin{cases}{{x}^{\mathrm{3}} −{x}}&{\left({x}\leq\mathrm{0}\right)}\\{{x}^{\mathrm{2}} }&{\left({x}\geq\mathrm{0}\right)}\end{cases}{dx} \\ $$
Answered by Frix last updated on 04/Nov/23
$$\underset{{a}} {\overset{{b}} {\int}}{f}\left({x}\right){dx}=\underset{{a}} {\overset{{c}} {\int}}{f}\left({x}\right){dx}+\underset{{c}} {\overset{{b}} {\int}}{f}\left({x}\right){dx} \\ $$$$\underset{−\mathrm{3}} {\overset{\mathrm{0}} {\int}}\left({x}^{\mathrm{3}} −{x}\right){dx}+\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}{x}^{\mathrm{2}} {dx}=−\frac{\mathrm{27}}{\mathrm{4}} \\ $$