True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: If $f(x, y, z)$ is a continuous function on the rectangular solid defined by $\mathcal{R}=\{(x, y, z)$ $a_{1} \leq x \leq a_{2}, b_{1} \leq y \leq b_{2},$ and $\left.c_{1} \leq z \leq c_{2}\right\},$ then
$\iiint_{\mathcal{R}} f(x, y, z) d V=\int_{a_{1}}^{a_{2}} \int_{b_{1}}^{b_{2}} \int_{c_{1}}^{c_{2}} f(x, y, z) d x d y d z$
(b) True or False: If $f(x, y, z)$ is a continuous function on the rectangular solid defined by $\mathcal{R}=\{(x, y, z)$ $a_{1} \leq x \leq a_{2}, b_{1} \leq y \leq b_{2},$ and $\left.c_{1} \leq z \leq c_{2}\right\},$ then
$\iiint_{\mathcal{X}} f(x, y, z) d V=\int_{a_{1}}^{a_{2}} \int_{c_{1}}^{c_{2}} \int_{b_{1}}^{b_{2}} f(x, y, z) d y d z d x$
(c) True or False: If $f(x, y, z)$ is a continuous function of three variables and $\Omega=\Omega_{1} \cup \Omega_{2}$ is a subset of $\mathbb{R}^{3},$ then $\iiint_{\Omega} f(x, y, z) d V=\iiint_{\Omega_{1}} f(x, y, z) d V+$
$\iiint_{\Omega_{2}} f(x, y, z) d V$
(d) True or False: If $\Omega$ is a bounded subset of $\mathbb{R}^{3}$, then there is a rectangular solid $\mathcal{R}$. with its sides parallel to the coordinate planes such that $\Omega \subseteq \mathcal{R}$
(e) True or False: If $f$ is a positive continuous function defined on a region $\Omega$ and $\Gamma \subseteq \Omega,$ then $\iiint_{\Gamma} f(x, y, z) d V \leq \iiint_{\Omega} f(x, y, z) d V$
(f) True or False: If $\rho(x, y, z)$ gives the density at every point of a region $\Omega,$ then the first moment of the mass in $\Omega$ with respect to the $x y$ -plane is $M_{x y}=$ $\iiint_{\Omega} x y \rho(x, y, z) d V$
(g) True or False: If $\rho(x, y, z)$ gives the density at every point of a region $\Omega,$ then the moment of inertia of $\Omega$ about the $x$ -axis is $I_{x}=\iiint_{\Omega}\left(y^{2}+z^{2}\right) \rho(x, y, z) d V$
(h) True or False: If $g_{1}(x, y) \leq g_{2}(x, y)$ on the square region $\mathcal{R}=\{(x, y) \mid 0 \leq x \leq 2$ and $0 \leq y \leq 2\},$ then the iter-
ated integral $\int_{g_{1}(x, y)}^{g_{2}(x, y)} \int_{0}^{2} \int_{0}^{2} d V$ represents the volume of the solid bounded below by $g_{1}(x, y)$ and above by
Double and Triple Integrals
Triple Integrals