Integration Questions

Question Number 54972 by peter frank last updated on 15/Feb/19

$$\int_{\mathrm{0}} ^{\pi} \int_{\mathrm{0}} ^{\mathrm{4cos}\:{z}} \int_{\mathrm{0}} ^{\sqrt{\mathrm{16}−{y}^{\mathrm{2}} }} {ydxdydz} \\$$

$$\left.\int_{\mathrm{0}} ^{\pi} \int_{\mathrm{0}} ^{\mathrm{4}{cosz}} {y}\:{x}\right]_{\mathrm{0}} ^{\sqrt{\mathrm{16}−{y}^{\mathrm{2}} }} {dydz}= \\$$$$\int_{\mathrm{0}} ^{\pi} \int_{\mathrm{0}} ^{\mathrm{4}{cosz}} \left({y}\sqrt{\mathrm{16}−{y}^{\mathrm{2}} }\right){dydz}= \\$$$$\left.\int_{\mathrm{0}} ^{\pi} \frac{−\mathrm{1}}{\mathrm{3}}\left(\mathrm{16}−{y}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} \right]_{\mathrm{0}} ^{\mathrm{4}{cosz}} {dz}= \\$$$$\frac{−\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\pi} \left(\left(\mathrm{16}−\mathrm{16}{cos}^{\mathrm{2}} {z}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} −\mathrm{16}^{\frac{\mathrm{3}}{\mathrm{2}}} \right){dz}= \\$$$$\frac{−\mathrm{16}^{\frac{\mathrm{3}}{\mathrm{2}}} }{\mathrm{3}}\int_{\mathrm{0}} ^{\pi} \left({sin}^{\mathrm{3}} {z}−\mathrm{1}\right){dz}=\frac{−\mathrm{16}^{\frac{\mathrm{3}}{\mathrm{2}}} }{\mathrm{3}}×\left(\frac{{sin}^{\mathrm{2}} {zcosz}+\mathrm{2}{cosz}}{\mathrm{3}}−{z}\right)_{\mathrm{0}} ^{\pi} = \\$$$$\frac{\mathrm{16}^{\frac{\mathrm{3}}{\mathrm{2}}} }{\mathrm{3}}×\left[\left(\frac{−\mathrm{2}}{\mathrm{3}}−\pi\right)−\left(\frac{\mathrm{2}}{\mathrm{3}}\right)\right]= \\$$$$\frac{−\mathrm{16}^{\frac{\mathrm{3}}{\mathrm{2}}} }{\mathrm{3}}×\left(\frac{−\mathrm{4}}{\mathrm{3}}−\pi\right)=\frac{\mathrm{256}}{\mathrm{9}}+\frac{\mathrm{64}}{\mathrm{3}}\pi \\$$$$\\$$$$\\$$