Evaluate the triple integral _E (xy)dV where E is a solid tetrahedron with vertices (0,0,0), (4,0,0), (0,1,0), (0,0,4)

To evaluate the triple integral, we need to determine the limits of integration for each variable x, y, and z.

The solid tetrahedron can be defined by the following three inequalities:

x ≥ 0
y ≥ 0
z ≥ 0

The tetrahedron is also bounded by the planes x + y + z ≤ 4, x + y ≤ 4, and y + z ≤ 4.

To set up the integral, we need to determine the limits of integration for each variable.

For z, the lower limit is 0, and the upper limit can be determined from the equation of the plane x + y + z = 4 as z = 4 - x - y. Therefore, the limits of integration for z are from 0 to 4 - x - y.

For y, the lower limit is 0, and the upper limit can be determined from the equation of the plane x + y = 4 as y = 4 - x. Therefore, the limits of integration for y are from 0 to 4 - x.

For x, the lower limit is 0, and the upper limit can be determined from the fact that one of the vertices is (4, 0, 0). Therefore, the limits of integration for x are from 0 to 4.

Now, we can set up the triple integral:

∫∫∫E (xy) dV

= ∫[0,4] ∫[0,4-x] ∫[0,4-x-y] (xy) dz dy dx

= ∫[0,4] ∫[0,4-x] [xy*z] evaluated from 0 to 4-x-y dy dx

= ∫[0,4] ∫[0,4-x] (xy(4-x-y) - xy(0)) dy dx

= ∫[0,4] ∫[0,4-x] (4xy - xyy - xy^2) dy dx

= ∫[0,4] (2x(4-x)^2 - x(4-x)^3/3 - x(4-x)^4/12) dx

Now, you can evaluate this integral using standard techniques of integration.