Q. 11 - Calculus

Question

Let $\rho (x, y,z)$ be a density function defined on the rectangular solid $R$ where $R=\left\{\right(x,y,z\left)\right|-1\le x\le 3,0\le y\le 2, and 2\le z\le 7\}$. Set up iterated integrals representing the mass of $R$, using all six distinct orders of integration.

Step-by-Step Solution

Answer

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

$$\begin{gathered} {\int }_{-1}^3{\int }_0^2{\int }_2^7 \rho \left(x,y,z\right)dzdydx \\ {\int }_0^2{\int }_{-1}^3{\int }_2^7 \rho \left(x,y,z\right)dzdxdy \\ {\int }_{-1}^3{\int }_2^7{\int }_0^2 \rho \left(x,y,z\right)dydzdx \\ {\int }_2^7{\int }_{-1}^3{\int }_0^2 \rho \left(x,y,z\right)dydxdz \\ {\int }_0^2{\int }_2^7{\int }_{-1}^3 \rho \left(x,y,z\right)dxdzdy \\ {\int }_2^7{\int }_0^2{\int }_{-1}^3 \rho \left(x,y,z\right)dxdydz \end{gathered}$$

Step 1 . Given information

$R=\left\{\right(x,y,z\left)\right|-1\le x\le 3,0\le y\le 2,and 2\le z\le 7\}$.

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 $R=\left\{\right(x,y,z\left)\right|a\le x\le b,c\le y\le d, and e\le z\le f\}$, 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 $R=\left\{\right(x,y,z\left)\right|-1\le x\le 3,0\le y\le 2, and 2\le z\le 7\}$, then the mass of the $R$ is defined as, $\underset{\Omega }{\iiint } \rho \left(x,y,z\right)dV={\int }_{-1}^3{\int }_0^2{\int }_2^7\rho \left(x,y,z\right)dzdydx$.

Step 3 . The other five distinct order of integration is,

$$\begin{gathered} {\int }_0^2{\int }_{-1}^3{\int }_2^7 \rho \left(x,y,z\right)dzdxdy \\ {\int }_{-1}^3{\int }_2^7{\int }_0^2 \rho \left(x,y,z\right)dydzdx \\ {\int }_2^7{\int }_{-1}^3{\int }_0^2 \rho \left(x,y,z\right)dydxdz \\ {\int }_0^2{\int }_2^7{\int }_{-1}^3 \rho \left(x,y,z\right)dxdzdy \\ {\int }_2^7{\int }_0^2{\int }_{-1}^3 \rho \left(x,y,z\right)dxdydz \end{gathered}$$