Find volume of Solid

Question

Find the volume of the solid bounded by x²+2y²=2, z=0, and x+y+2z=2

Answer 1

x+y+2z=2 becomes z = 1 - x/2 - y/2 and is a plane. At x=y=0, z=1

 

Note that this means the solid is just half of a right elliptic cylinder of height h=2 (the plane splits the cylinder into two congruent parts).

 

The base of the cylinder has semiaxes a=1/√2, b=1 and area A = πab = π/√2

 

The volume of the solid is thus V = πabh/2 = π/√2

 

You can also get this with calculus with a triple integral, but it would be a bit messy.