Final answer:
The volume of the region in the first octant bounded by the coordinate planes and the plane x + y + z = 3 is 2.25 cubic units. The region forms a tetrahedron, and its volume can be found using the formula for the volume of a tetrahedron.
Step-by-step explanation:
To find the volume of the region in the first octant bounded by the coordinate planes and the plane described by the equation x + y + z = 3, we can integrate in rectangular coordinates or use geometric reasoning. Using geometric reasoning, we note that the region is a tetrahedron with edges 3 units long along the x, y, and z-axis, respectively. Because the region is bounded by the coordinate planes, the vertices of the tetrahedron are (0,0,3), (0,3,0), and (3,0,0), with the origin as the fourth vertex.
To determine the volume of a tetrahedron, we can use the formula V = (1/6) * base area * height. In this case, the base is a right triangle in the xy-plane with sides of length 3, so the area is (1/2) * 3 * 3 = 4.5 square units. The height is 3 units (along the z-axis). Plugging these values into the volume formula yields V = (1/6) * 4.5 * 3 = 2.25 cubic units.
Therefore, the volume of the tetrahedron is 2.25 cubic units.