Q. 13 - Calculus

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.

Step-by-Step Solution

Answer

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

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

Step 1 . Given information

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 $(0, 0,3)$.

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 $y$ the limits from oblique plane becomes,

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

Step 3 . Similarly the other four mass integrals are,

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