Question Number 203186 by lorance last updated on 11/Jan/24
$${f}\left({x}\right)=\left\{_{\mathrm{2}\:\:\:\:\:\:\:\:{x}=\mathrm{1}} ^{\mathrm{7}\:\:\:\:\:\:\:\:{x}\neq\mathrm{1}\:\:\:\:\:} \Rightarrow\:\int_{\mathrm{0}} ^{\:\mathrm{4}} {f}\left({x}\right){dx}=?\right. \\ $$
Answered by mr W last updated on 12/Jan/24
$$\int_{\mathrm{0}} ^{\mathrm{4}} {f}\left({x}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}+\int_{\mathrm{1}} ^{\mathrm{4}} {f}\left({x}\right){dx} \\ $$$$=\underset{{a}\rightarrow\mathrm{1}^{−} } {\mathrm{lim}}\int_{\mathrm{0}} ^{{a}} {f}\left({x}\right){dx}+\underset{{b}\rightarrow\mathrm{1}^{+} } {\mathrm{lim}}\int_{{b}} ^{\mathrm{4}} {f}\left({x}\right){dx} \\ $$$$=\underset{{a}\rightarrow\mathrm{1}^{−} } {\mathrm{lim}7}×\left({a}−\mathrm{0}\right)+\underset{{b}\rightarrow\mathrm{1}^{+} } {\mathrm{lim}7}×\left(\mathrm{4}−{b}\right) \\ $$$$=\mathrm{7}×\left(\mathrm{1}−\mathrm{0}\right)+\mathrm{7}×\left(\mathrm{4}−\mathrm{1}\right) \\ $$$$=\mathrm{7}×\left(\mathrm{4}−\mathrm{0}\right) \\ $$$$=\mathrm{28} \\ $$