Find limits of ly, uy, lx, ux, lz, uz so that ∫uzlz∫uxlx∫uylydydxdz represents the volume of the region in the first octant that is bounded by the 3 coordinate planes and the plane x+y+3z=3.

Answer 1

∫₀¹dz∫₀^3–3zdx∫₀^3–x–3zdy but to see why this is, one really needs to draw a good picture.

l(x,z) = ∫₀^3–x–3zdy is the length of a horizontal pin from (x,0,z) to (x,3–x–3z,z).

a(z) = ∫₀^3–3zdxl(x,z) is the area of a horizontal triangle (made up of many pins of areas dxl(x,z)) at height z between x = 0, y = 0 and x = 3–3z.

v = ∫₀¹dza(z) is the volume (made up of many triangles of volumes dza(z)).