Question

Let $\rho (x,y,z)$be a density function defined on the rectangular solid $R$where role="math" localid="1650360487967" $R=\left\{\right(x,y,z\left)\right|a_1\le x\le a_2,b_1\le y\le b_2,and{c}_1\le z\le c_2\}$. Set up iterated integrals representing the mass of $R$, using all six distinct orders of integration.

Short answer

The six distinct orders of integration representing the mass of $R$are,

role="math" localid="1650361542302" $$\begin{gathered} {\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}\rho \left(x,y,z\right)dzdydx \\ {\int }_{b_1}^{b_2}{\int }_{a_1}^{a_2}{\int }_{c_1}^{c_2}\rho \left(x,y,z\right)dzdxdy \\ {\int }_{a_1}^{a_2}{\int }_{c_1}^{c_2}{\int }_{b_1}^{b_2}\rho \left(x,y,z\right)dydzdx \\ {\int }_{c_1}^{c_2}{\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}\rho \left(x,y,z\right)dydxdz \\ {\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}{\int }_{a_1}^{a_2}\rho \left(x,y,z\right)dxdzdy \\ {\int }_{c_1}^{c_2}{\int }_{b_1}^{b_2}{\int }_{a_1}^{a_2}\rho \left(x,y,z\right)dxdydz \end{gathered}$$

Step-by-step solution

Step 1. Given information

$R=\left\{\right(x,y,z\left)\right|a_1\le x\le a_2,b_1\le y\le b_2,and{c}_1\le z\le c_2\}$.

Step 2. The definition of the iterated triple integral:

If $\rho \left(x,y,z\right)$be a density function defined on the rectangular solid $R$where role="math" localid="1650361021093" $R=\left\{\right(x,y,z\left)\right|a\le x\le b,c\le y\le d,ande\le z\le f\}$then the mass of $R$is defined as, $\underset{\Omega }{\iiint }\rho \left(x,y,z\right)dV={\int }_a^b{\int }_c^d{\int }_e^f\rho \left(x,y,z\right)dzdydx$.

The other mass equations are defined in the same way for other $5$triple integrals. Given that $\rho \left(x,y,z\right)$be a density function defined on the rectangular solid $R$where $R=\left\{\right(x,y,z\left)\right|a_1\le x\le a_2,b_1\le y\le b_2,and{c}_1\le z\le c_2\}$then the mass of $R$is defined as role="math" localid="1650361522159" $\underset{\Omega }{\iiint }\rho \left(x,y,z\right)dV={\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}\rho \left(x,y,z\right)dzdydx$.

Step 3. The other five equations for the mass are,

$$\begin{gathered} {\int }_{b_1}^{b_2}{\int }_{a_1}^{a_2}{\int }_{c_1}^{c_2}\rho \left(x,y,z\right)dzdxdy \\ {\int }_{a_1}^{a_2}{\int }_{c_1}^{c_2}{\int }_{b_1}^{b_2}\rho \left(x,y,z\right)dydzdx \\ {\int }_{c_1}^{c_2}{\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}\rho \left(x,y,z\right)dydxdz \\ {\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}{\int }_{a_1}^{a_2}\rho \left(x,y,z\right)dxdzdy \\ {\int }_{c_1}^{c_2}{\int }_{b_1}^{b_2}{\int }_{a_1}^{a_2}\rho \left(x,y,z\right)dxdydz \end{gathered}$$