Question

Let $f\left(x\right)$be an integrable function on the rectangular solid $\mathfrak{R}=\left\{\left(x,y,z\right)|a_1⩽x⩽a_2,b_1⩽y⩽b_2,c_1⩽z⩽c_2\right\}$, and let $\kappa \in \mathrm{ℝ}.$Use the definition of the triple integral to prove that:

${\iiint }_{\mathfrak{R}}\kappa f(x,y,z)dV=\kappa {\iiint }_{\mathfrak{R}}f(x,y,z)dV.$

Short answer

$$\begin{gathered} {\iiint }_{\mathfrak{R}}f(x,y,z)dV=\underset{∆\to 0}{\lim }\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i^{*},y_j^{*},z_k^{*})dV. \\ Replace\kappa f(x,y,z)withf(x,y,z)weget, \\ {\iiint }_{\mathfrak{R}}\kappa f(x,y,z)dV=\kappa \underset{∆\to 0}{\lim }\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i^{*},y_j^{*},z_k^{*})dV=\kappa \left({\iiint }_{\mathfrak{R}}f(x,y,z)dV\right) \\ Therefore, \\ {\iiint }_{\mathfrak{R}}\kappa f(x,y,z)dV=\kappa {\iiint }_{\mathfrak{R}}f(x,y,z)dV. \end{gathered}$$

Step-by-step solution

Step 1. Given Information.

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

Step 2. Proof.

$$\begin{gathered} Asweknowfromthedefinitionoftripleintegrals: \\ {\iiint }_{\mathfrak{R}}f(x,y,z)dV=\underset{∆\to 0}{\lim }\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i^{*},y_j^{*},z_k^{*})dV. \\ Replace\kappa f(x,y,z)withf(x,y,z)inabove, \\ {\iiint }_{\mathfrak{R}}\kappa f(x,y,z)dV=\underset{∆\to 0}{\lim }\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^n\kappa f(x_i^{*},y_j^{*},z_k^{*})dV \\ Asweknowthecons\tan tsgetoutfromsummationsandlimitsweget, \\ {\iiint }_{\mathfrak{R}}\kappa f(x,y,z)dV=\kappa \underset{∆\to 0}{\lim }\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i^{*},y_j^{*},z_k^{*})dV \\ =\kappa \left(\underset{∆\to 0}{\lim }\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i^{*},y_j^{*},z_k^{*})dV\right) \\ =\kappa \left({\iiint }_{\mathfrak{R}}f(x,y,z)dV\right) \\ Therefore, \\ {\iiint }_{\mathfrak{R}}\kappa f(x,y,z)dV=\kappa {\iiint }_{\mathfrak{R}}f(x,y,z)dV. \end{gathered}$$