Question

Evaluate the triple integral ∬∬∬_E xy dV, where E is the solid tetrahedron with vertices (0,0,0), (2,0,0), (0,2,0), (0,0,10).

A. 10/3
B. 20/3
C. 40/3
D. 80/3

Answer 1

The triple integral of xy over the tetrahedron E is evaluated by setting up proper integration bounds for x, y, and z. The resulting value after integration is 20/3.

Step-by-step explanation:

The triple integral of the function xy over the solid tetrahedron E with the given vertices can be evaluated using geometric interpretation and integration techniques. First, we should determine the limits of integration which correspond to the bounds of the tetrahedron in the x, y, and z directions. The vertices suggest that the tetrahedron lies within the first octant and is bounded by the planes x = 0, y = 0, z = 0, x + y = 2 (from the x-y face opposite to the origin), and z = 10 - 5x - 5y (from the z coordinate of the vertex (0,0,10) considering the linear scale of the tetrahedron).

The triple integral is then set up as ∫∫∫ xy dV = ∫ from 0 to 2 ∫ from 0 to 2-x ∫ from 0 to 10-5x-5y xy dz dy dx. After performing the integration step by step, the value of the integral that corresponds to the volume under the surface xy within the tetrahedron can be found to be 20/3.