Question Number 207169 by hardmath last updated on 08/May/24
$$\mathrm{Find}:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{3}} \:\sqrt{\mathrm{9}\:−\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$
Answered by lepuissantcedricjunior last updated on 08/May/24
$$\int_{\mathrm{0}} ^{\mathrm{3}} \sqrt{\mathrm{9}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\boldsymbol{\mathrm{dx}}=\frac{\boldsymbol{\pi}\mathrm{3}^{\mathrm{2}} }{\mathrm{4}}=\frac{\mathrm{9}}{\mathrm{4}}\boldsymbol{\pi} \\ $$$$\boldsymbol{{car}}\:\forall\boldsymbol{{a}}\in\mathbb{R}_{+} ^{\ast} \int_{\mathrm{0}} ^{\boldsymbol{{a}}} \sqrt{\boldsymbol{{a}}^{\mathrm{2}} −\boldsymbol{{x}}^{\mathrm{2}} }\boldsymbol{{dx}}=\frac{\boldsymbol{{a}}^{\mathrm{2}} }{\mathrm{4}}\boldsymbol{\pi} \\ $$$$……….. \\ $$