Final Answer:
The triple integral to find the volume (V) of the tetrahedron enclosed by the coordinate planes and the plane 10x + y + z = 4 is given by:
V = ∫₀⁴ ∫₀^(4-y) ∫₀^(4-y-z) dz dx dy
Step-by-step explanation:
To set up the triple integral, we need to define the limits of integration for each variable. The integral is evaluated over the region enclosed by the coordinate planes (x, y, z) and the plane 10x + y + z = 4. The limits for z are from 0 to 4-y, as z is restricted by the plane. Similarly, the limits for x are from 0 to 4-y, and for y, the limits are from 0 to 4. The integral represents the volume of the tetrahedron as the sum of infinitesimally small volumes.
In detail, the innermost integral dz is integrated from 0 to 4-y-z, representing the distance between the plane and the point (x, y, z). The middle integral dx is integrated over the range of x values, and the outermost integral dy is integrated over the entire range of y values. This setup captures the volume enclosed by the coordinate planes and the given plane.
The triple integral provides a systematic way to compute the volume of the tetrahedron, utilizing the limits to consider each dimension and the relationships between them.