5.6 Triple integrals

(a) $\displaystyle{abc}$ (b) $\displaystyle{\frac{1}{12}}$

(a) $\displaystyle{\int_{\theta = 0}^{2\pi} \int_{\phi = 0}^{\pi} \int_{\rho =0}^{a} k\rho^{3}\sin{\phi}d\rho d\phi d\theta = k\pi a^{4}}$

(b) $\displaystyle{\int_{\theta = 0}^{2\pi} \int_{\phi = 0}^{\pi/4} \int_{\rho =0}^{... ...} k\rho^{3}\sin{\phi}d\rho d\phi d\theta = \frac{k\pi}{6}(2^{\frac{3}{2}} - 1)}$

(c) $\displaystyle{\iint_{\Omega}\int_{z=2+y^{2}}^{4-x^{2}-y^{2}}dz dx dy = \int_{0}^{2\pi}\int_{0}^{1}(2 - 2r^{2})\sqrt{2}rdr d\theta = \sqrt{2}\pi}$

(a) $\displaystyle{(\bar{x}, \bar{y}) = (\frac{1}{2}, \frac{2}{5})}$ (b) $\displaystyle{(\bar{x}, \bar{y}) = (1, \frac{8}{5})}$ (c) $\displaystyle{(\bar{x}, \bar{y}) = (2,4)}$