Find limits ly, uy, lx, ux, lz, uz so that ∫uyly∫uzlz∫uxlxdxdzdy represents the volume of the region in the first octant that is inside the paraboloid y=x2+8z2 between y=8, y=10.

Answer 1

l(y,z) = ∫₀^{√(y–8z²)}dx is the length of a horizontal pin between (0,y,z) on the x=0 plane and (√(y–8z²),y,z) on the paraboloid.

a(y) = ∫₀^√(y/8)l(y,z)dz is the area of many such pins stacked into a vertical elliptical quarter-disk.

∫₈¹⁰a(y)dy is the volume of many such quarter-disks laid out horizontally.