Question

Find the average value of the function over the given solid. The average value of a continuous function f(x, y, z) over a solid region Q is 1 V Q f(x, y, z) dV where V is the volume of the solid region Q. f(x, y, z) = x + y + z over the tetrahedron in the first octant with vertices (0, 0, 0), (2, 0, 0), (0, 2, 0) and (0, 0, 2).

Answer 1

Compute the volume of $Q$:

$\displaystyle\iiint_Q\mathrm dV=\int_0^2\int_0^{2-x}\int_0^{2-x-y}\mathrm dz\,\mathrm dy\,\mathrm dx=\frac43$

Integrate $f(x,y,z)=x+y+z$ over $Q$:

$\displaystyle\iiint_Qf(x,y,z)\,\mathrm dV=\int_0^2\int_0^{2-x}\int_0^{2-x-y}(x+y+z)\,\mathrm dz\,\mathrm dy\,\mathrm dx=2$

So the average value of $f$ over $Q$ is 2/(4/3) = 3/2.