Question Number 201762 by Calculusboy last updated on 11/Dec/23
Answered by mr W last updated on 12/Dec/23
$$\int_{−\mathrm{2}} ^{\mathrm{2}} \left({x}^{\mathrm{3}} \mathrm{cos}\:\frac{{x}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }\:{dx} \\ $$$$=\int_{−\mathrm{2}} ^{\mathrm{2}} \underset{{odd}\:{function}} {\left({x}^{\mathrm{3}} \mathrm{cos}\:\frac{{x}}{\mathrm{2}}\right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }\:}{dx}+\int_{−\mathrm{2}} ^{\mathrm{2}} \underset{{even}\:{function}} {\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }\:}{dx} \\ $$$$=\mathrm{0}+\int_{\mathrm{0}} ^{\mathrm{2}} \sqrt{\mathrm{4}−{x}^{\mathrm{2}} }\:{dx} \\ $$$$=\frac{\pi×\mathrm{2}^{\mathrm{2}} }{\mathrm{4}}=\pi\:\checkmark \\ $$
Commented by Calculusboy last updated on 11/Dec/23
$$\boldsymbol{{thanks}}\:\boldsymbol{{sir}} \\ $$
Commented by mr W last updated on 12/Dec/23
$${if}\:{f}\left(−{x}\right)=−{f}\left({x}\right),\:{then} \\ $$$$\int_{−{a}} ^{{a}} {f}\left({x}\right){dx}=\mathrm{0}. \\ $$$${if}\:{f}\left(−{x}\right)={f}\left({x}\right),\:{then} \\ $$$$\int_{−{a}} ^{{a}} {f}\left({x}\right){dx}=\mathrm{2}\int_{\mathrm{0}} ^{{a}} {f}\left({x}\right){dx} \\ $$