Question Number 167588 by AlbertEinstein last updated on 20/Mar/22 | ||
$$\mathrm{1}+\mathrm{1}=¿ \\ $$ | ||
Answered by chhaythean last updated on 20/Mar/22 | ||
$$=\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\mathrm{2}\pi} \int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \mathrm{sin}\varphi\mathrm{cos}^{\mathrm{3}} \theta\mathrm{d}\varphi\mathrm{d}\theta \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\mathrm{2}\pi} \mathrm{cos}^{\mathrm{3}} \theta×\left[−\mathrm{cos}\varphi\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \mathrm{d}\theta \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}×\left[−\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}−\mathrm{1}\right)\right]\left(\int_{\mathrm{0}} ^{\mathrm{2}\pi} \mathrm{cos}\theta\mathrm{d}\theta−\int_{\mathrm{0}} ^{\mathrm{2}\pi} \mathrm{sin}^{\mathrm{2}} \theta\mathrm{d}\left(\mathrm{sin}\theta\right)\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}×\left[−\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}−\mathrm{1}\right)\right]×\mathrm{0} \\ $$$$\mathrm{So}\:\begin{array}{|c|}{\int_{\mathrm{0}} ^{\mathrm{2}\pi} \int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \int_{\mathrm{0}} ^{\mathrm{cos}\theta} \mathrm{r}^{\mathrm{2}} \mathrm{sin}\varphi\mathrm{drd}\varphi\mathrm{d}\theta=\mathrm{0}}\\\hline\end{array} \\ $$ | ||