Find the volume

Question

Find the volume of the solid region bounded in the first octant by the plane given by the equation z + x + 2y = 6

Answer 1

A non-calculus approach is to note that the three points

(6,0,0) , (0,3, 0) and (0, 0, 6) define the triangular region in the plane that, together with the boundaries of the first octant, enclose the volume which is to be calculated.

The points (0, 0, 0), (6, 0 , 0) and ( 0 , 3, 0) define a triangle in the x,y plane of area (1/2) 6 x 3 = 9

This triangle can be thought of as the base of triangle base pyramid with height 6.

A formula for the volume of such a pyramid is (1/3) height x area-of-base.

This works out to be (1/3) 6 x 9 = 18. This is the desired volume.