Question

find the volume of the solid bounded above by the surface z=f(x,y) and below by the plane region R
f(x,y)= 4-2x-y; R{(x,y)0<x<1;0<y<2

Answer 1

this is pretty straightforward. Nice rectangular base.

v = ∫∫R f(x,y) dA
= ∫[0,1]∫[0,2] 4-2x-y dy dx
= ∫[0,1] (4y - 2xy - 1/2 y^2)[0,2] dx
= ∫[0,1] (6-4x) dx
= (6x - 2x^2)[0,1]
= 4