Question

Evaluate the triple integral ///(E) xydV where E is the solid

tetrahedon with vertices (0,0,0),(3,0,0),(0,5,0),(0,0,4).

Answer 1

The value of the triple integral
`\( \iiint_E xy \, dV \)` over the tetrahedron with vertices (0,0,0), (3,0,0), (0,5,0), and (0,0,4) is 15.

Step-by-step explanation:

To evaluate the triple integral
`\( \iiint_E xy \, dV \)` over the given tetrahedron, we first need to understand the limits of integration for each variable (x, y, and z). The given tetrahedron has vertices at (0,0,0), (3,0,0), (0,5,0), and (0,0,4), which form a solid tetrahedron in 3D space.

The limits for x range from 0 to 3, the limits for y range from 0 to 5, and the limits for z range from 0 to 4, as these values bound the solid tetrahedron in each direction.

The integrand is xy , which represents the product of x and y. Setting up and solving the triple integral in the given limits, the evaluation involves integrating xy over the specified region in xyz-space.

`\[\iiint_E xy \, dV = \int_0^4 \int_0^(5-z) \int_0^(3) xy \, dx \, dy \, dz\]`

Integrating xy with respect to x from 0 to 3, y from 0 to (5-z), and z from 0 to 4, and performing the calculations, the final value of the triple integral is 15. This represents the volume integral of the function xy over the given tetrahedron bounded by the specified vertices.