5.6 Triple integrals

1.

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

2.

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

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

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

3.

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