Final answer:
D. (V = iint dV)
To find the volume of a tetrahedron bounded by the coordinate planes and a plane, we use a triple integral with limits of integration based on the intercepts a, b, and c. The correct choice is D. (V = iint dV), which signifies a triple integral over a volume.
Step-by-step explanation:
The volume of a tetrahedron bounded by the coordinate planes and a plane in three-dimensional space can be computed using a triple integral. The points (a,0,0), (0,b,0), and (0,0,c) define a plane that cuts through the coordinate axes, forming a tetrahedron. To find the volume, V, of this tetrahedron, we set up a triple integral in Cartesian coordinates (where dV = dx dy dz), with limits of integration determined by the intercepts a, b, and c on the x-, y-, and z-axes respectively.
We realize that as we move along the x-axis from 0 to a, the corresponding y-value ranges from 0 to a linear function decreasing from b to 0, and similarly for the z-value. Therefore, the limits of integration for y will be from 0 to b(1-x/a) and for z from 0 to c(1-x/a)(1-y/b). The triple integral then becomes:
V = ∫0a ∫0b(1-x/a) ∫0c(1-x/a)(1-y/b) dz dy dx
When this integral is evaluated, it simplifies to V = (1/6)abc, which is a known formula for the volume of a tetrahedron, thus confirming that option D. (V = iint dV) represents the correct approach to use a triple integral to find the volume of the tetrahedron.