Question

Let E be the solid tetrahedron with vertices on (6, 1, 0), (0, 1, 0), (0, 1, 2) and (0, 4, 0). Write the triple integral f(x, y, z) dV as an iterated integral in the order dy dz dx​

Answer 1

Observe that
`x=0`,
`y=1` and
`z=0` are common to 3 of the 4 given points. This tells us that tetrahedron has faces in the planes
`x=0`,
`y=1`, and
`z=0`.

Find the equation of the last plane that closes off the tetrahedron. This plane passes through the points (6, 1, 0), (0, 1, 2), and (0, 4, 0). Find two vectors that run parallel to this plane.

`\vec v_1 = \langle6,1,0\rangle - \langle0,1,2\rangle = \langle6,0,-2\rangle`

`\vec v_2 = \langle6,1,0\rangle - \langle0,4,0\rangle = \langle6,-3,0\rangle`

Compute their cross product to get a vector that is perpendicular to the plane.

`\vec n = \vec v_1 * \vec v_2 = \langle -6, -12, -18\rangle`

Then using any of the 3 points it contains, we find the equation of the plane to be e.g.

`\vec n \cdot \langle x-6,y-1,z-0\rangle = 0 \implies x+2y+3z=8`

Now we can set up the integration region.
`E` is the set

`E = \left\{(x,y,z) \mid 1 \le y \le \frac{8 - x - 3z}2 \text{ and } 0 \le z \le \frac{6 - x}3 \text{ and } 0 \le x \le 6\right\}`

so that the integral taken in the prescribed order of variables is

`\displaystyle \iiint_E f(x,y,z) \, dV = \boxed{\int_0^6 \int_0^((6-x)/3) \int_1^((8-x-3z)/2) f(x,y,z) \, dy \, dz \, dx}`