Question

$$\begin{gathered} Let\mathfrak{R}=\left\{\left(x,y,z\right)|a_1⩽x⩽a_2,b_1⩽y⩽b_2,c_1⩽z⩽c_2\right\}.Provethat: \\ {\iiint }_{\mathfrak{R}}dV=\left(a_2-a_1\right)\left(b_2-b_1\right)\left(c_2-c_1\right). \\ Whatistherelationshipbetween\mathfrak{R}andtheproduct\left(a_2-a_1\right)\left(b_2-b_1\right)\left(c_2-c_1\right). \end{gathered}$$

Short answer

$$\begin{gathered} {\iiint }_{\mathfrak{R}}dV={\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}dzdydx \\ ={\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}\left[{\int }_{c_1}^{c_2}dz\right]dydx \\ =\left(c_2-c_1\right){\int }_{a_1}^{a_2}\left[{\int }_{b_1}^{b_2}dy\right]dx \\ =\left(c_2-c_1\right)\left(b_2-b_1\right){\int }_{a_1}^{a_2}dx \\ =\left(a_2-a_1\right)\left(b_2-b_1\right)\left(c_2-c_1\right) \\ {\iiint }_{\mathfrak{R}}dV=\left(a_2-a_1\right)\left(b_2-b_1\right)\left(c_2-c_1\right). \end{gathered}$$

Step-by-step solution

Step 1. Given Information.

$Given:\mathfrak{R}=\left\{\left(x,y,z\right)|a_1⩽x⩽a_2,b_1⩽y⩽b_2,c_1⩽z⩽c_2\right\}$

Step 2. Proof.

$$\begin{gathered} Thetripleintegralisevaluatedasfollows: \\ {\iiint }_{\mathfrak{R}}dV={\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}dzdydx \\ ={\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}\left[{\int }_{c_1}^{c_2}dz\right]dydx \\ ={\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}\left(c_2-c_1\right)dydx \\ =\left(c_2-c_1\right){\int }_{a_1}^{a_2}\left[{\int }_{b_1}^{b_2}dy\right]dx \\ =\left(c_2-c_1\right){\int }_{a_1}^{a_2}\left(b_2-b_1\right)dx \\ =\left(c_2-c_1\right)\left(b_2-b_1\right){\int }_{a_1}^{a_2}dx \\ =\left(c_2-c_1\right)\left(b_2-b_1\right)\left(a_2-a_1\right) \\ =\left(a_2-a_1\right)\left(b_2-b_1\right)\left(c_2-c_1\right) \\ =RHS. \\ Hence,proved. \end{gathered}$$