Q. 14 - Calculus

Question

Let $\rho (x, y,z)$ be a density function defined on the tetrahedron $\Omega$ with vertices $(0, 0, 0), (a, 0, 0), (0, b, 0),$ and $(0, 0,c),$ where $a, b,$and $c$ are positive real numbers. Set up iterated integrals representing the mass of $\Omega$ , using all six distinct orders of integration.

Step-by-Step Solution

Answer

The five distinct orders of integration representing the mass of $\Omega$ is,

$$\begin{gathered} {\int }_0^b{\int }_0^{c-\frac{y}{b}}{\int }_0^{c-\frac{cx}{a}-\frac{cy}{a}} \rho \left(x,y,z\right)dzdxdy \\ {\int }_0^a{\int }_0^{b-\frac{cx}{a}}{\int }_0^{b-\frac{bx}{a}-\frac{bz}{c}} \rho \left(x,y,z\right)dydzdx \\ {\int }_0^c{\int }_0^{b-\frac{bz}{c}}{\int }_0^{b-\frac{bx}{a}-\frac{bz}{c}} \rho \left(x,y,z\right)dydxdz \\ {\int }_0^c{\int }_0^{b-\frac{bz}{c}}{\int }_0^{a-\frac{ay}{b}-\frac{az}{c}} \rho \left(x,y,z\right)dxdydz \\ {\int }_0^b{\int }_0^{c-\frac{cy}{b}}{\int }_0^{a-\frac{ay}{b}-\frac{az}{c}} \rho \left(x,y,z\right)dxdzdy \end{gathered}$$

Step 1 . Given information

Let $\rho (x, y,z)$ be a density function defined on the tetrahedron  with vertices $(0, 0, 0), (a, 0, 0), (0, b, 0),$ and $(0, 0,c)$.

Step 2 . Similarly the other five mass integrals are,

${\int }_0^b{\int }_0^{c-\frac{y}{b}}{\int }_0^{c-\frac{cx}{a}-\frac{cy}{a}} \rho \left(x,y,z\right)dzdxdy$.

Since the triangular region becomes ${\Omega }_{yz}$ then the ordinate becomes $0\le y\le b,0\le x\le c-\frac{c}{a}x and 0\le z\le c-\frac{cx}{a}-\frac{cy}{a}$.

The other four mass equations are,

$$\begin{gathered} {\int }_0^a{\int }_0^{b-\frac{cx}{a}}{\int }_0^{b-\frac{bx}{a}-\frac{bz}{c}} \rho \left(x,y,z\right)dydzdx \\ {\int }_0^c{\int }_0^{b-\frac{bz}{c}}{\int }_0^{b-\frac{bx}{a}-\frac{bz}{c}} \rho \left(x,y,z\right)dydxdz \\ {\int }_0^c{\int }_0^{b-\frac{bz}{c}}{\int }_0^{a-\frac{ay}{b}-\frac{az}{c}} \rho \left(x,y,z\right)dxdydz \\ {\int }_0^b{\int }_0^{c-\frac{cy}{b}}{\int }_0^{a-\frac{ay}{b}-\frac{az}{c}} \rho \left(x,y,z\right)dxdzdy \end{gathered}$$