Question

Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 6.

Answer 1

36

Answer 2

x+y+z=6 is a plane crossing all axes at their respective values 6. The shape bound by these four is a tetrahedron with equilateral base (length 6sqrt2). Base, therefore, has area 18sqrt3.Height is, I hope, sqrt(141/8) so, volume = 1/3*height*area of base = 4.5sqrt94

Answer 3

think of it as one quarter of a pyramid. the total pyramid the base has lengths \[6\sqrt{2}\]

so the area of the base is \[(6\sqrt{2})^2=72\] the height of the pyramid is 6

so the volume is give by the formula \[V=\frac{1}{3}Bh=\frac{1}{3}72\cdot 6\]

but we only want one quarter of it so \[\frac{1}{3}72\cdot 6/4=36\]


or you can use integration

\[V=\int\limits_{0}^{6}\int\limits_{0}^{6-x}(6-x-y)\,dy\,dx=36\]