Question

Let $\rho (x,y,z)$be a density function defined on the tetrahedron $\Omega$with vertices $(0,0,0),(2,0,0),(0,4,0),$and $(0,0,3)$. Set up iterated integrals representing the mass of , using all six distinct orders of integration.

Short answer

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

role="math" localid="1650366865545" $$\begin{gathered} {\int }_0^1{\int }_0^{3-\frac{3x}{2}}{\int }_0^{4-2x-\frac{4y}{3}}\rho \left(x,y,z\right)dydzdx \\ {\int }_0^3{\int }_0^{2-\frac{2x}{3}}{\int }_0^{4-2x-\frac{4y}{3}}\rho \left(x,y,z\right)dydxdz \\ {\int }_0^4{\int }_0^{3-\frac{3y}{4}}{\int }_0^{2-\frac{y}{2}-\frac{2z}{3}}\rho \left(x,y,z\right)dxdzdy \\ {\int }_0^3{\int }_0^{4-\frac{4z}{3}}{\int }_0^{2-\frac{y}{2}-\frac{2z}{3}}\rho \left(x,y,z\right)dxdydz \\ {\int }_0^4{\int }_0^{2-\frac{y}{2}}{\int }_0^{3-\frac{3x}{2}-\frac{3y}{4}}\rho \left(x,y,z\right)dzdxdy \end{gathered}$$

Step-by-step solution

Step 1. Given information

Let $\rho (x,y,z)$be a density function defined on the tetrahedron$\Omega$with vertices localid="1650434213243" $(0,0,0),(2,0,0),(0,4,0),$and localid="1650434285733" $(0,0,3)$.

Step 2. The other five mass integrals are,

${\int }_0^4{\int }_0^{2-\frac{y}{2}}{\int }_0^{3-\frac{3x}{2}-\frac{3y}{4}}\rho \left(x,y,z\right)dzdxdy$.

Since taking $z$and $x$after $y$thelimits from oblique plane becomes,

$0\le y\le 4,0\le x\le 2-\frac{y}{2}and0\le z\le 1-\frac{3x}{2}-\frac{3y}{4}$.

Step 3. Similarly the other four mass integrals are,

$$\begin{gathered} {\int }_0^1{\int }_0^{3-\frac{3x}{2}}{\int }_0^{4-2x-\frac{4y}{3}}\rho \left(x,y,z\right)dydzdx \\ {\int }_0^3{\int }_0^{2-\frac{2x}{3}}{\int }_0^{4-2x-\frac{4y}{3}}\rho \left(x,y,z\right)dydxdz \\ {\int }_0^4{\int }_0^{3-\frac{3y}{4}}{\int }_0^{2-\frac{y}{2}-\frac{2z}{3}}\rho \left(x,y,z\right)dxdzdy \\ {\int }_0^3{\int }_0^{4-\frac{4z}{3}}{\int }_0^{2-\frac{y}{2}-\frac{2z}{3}}\rho \left(x,y,z\right)dxdydz \end{gathered}$$