Question

Evaluate the triple integral. t 5x2 dv, where t is the solid tetrahedron with verticies (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1)

Answer 1

The face of the tetrahedron in the first octant, not coinciding with any of the planes generated by the coordinate axes, is given by the plane $x+y+z=0$. So the integral can be set up as

$\displaystyle\iiint_{\mathcal T}5x^2\,\mathrm dV=\int_{x=0}^{x=1}\int_{y=0}^{y=1-x}\int_{z=0}^{z=1-x-y}5x^2\,\mathrm dz\,\mathrm dy\,\mathrm dx=\frac1{12}$