Back

Multiple Integral Fubini

A double integral is the volume of the solid trapped between a surface and a patch of the plane. You cannot add up a whole volume at once, so you slice the solid into thin slabs and add the slabs. Cut perpendicular to x and each slab is the area under the surface along a strip, the inner integral over y; stack those slabs along x and you have the volume. Cut perpendicular to y instead and you get a different set of slabs, stacked the other way. These are the two iterated integrals. Fubini's theorem says that for a well-behaved surface it does not matter which way you slice the loaf: both stacks fill the same solid, so both orders give exactly the same number. That is the permission slip behind half of multivariable calculus, letting you swap the order whenever one inner integral is easier. The scene shows the solid in 3D, sliced in the order you pick, with a sweep that fills the slabs in one at a time. The diagnostic accumulates the volume for both orders at once: two running totals that take different routes and land on precisely the same value.

Figure 1. Fubini's theorem as a sliceable volume. Top: the solid under z = f(x,y) over a resizable region, cut into slabs along x (dy then dx) or along y (dx then dy), with a sweep filling the slabs in. Bottom: the volume accumulated in both orders, taking different paths but ending at the same total. Method: nested Simpson quadrature of the iterated integrals; oblique Canvas2D projection of the solid.
integrandsin x · sin y
slicedy dx
regiondrag the back corner

WHAT TO TRY

  • Watch the slabs fill in as the sweep advances and the running volume climbs to V.
  • Switch the slice order: the solid is now cut the other way (walls front-to-back instead of fins left-to-right), yet the final volume is identical. That is Fubini.
  • Pick the slant integrand: now the two accumulation curves take visibly different routes (one climbs like x², the other like a sine) before meeting at the same V. The dome is symmetric, so its two routes coincide.
  • The wave is not separable (it cannot be split into a function of x times a function of y), so the freedom to swap the order is genuine, not a trivial factoring.
  • Drag the back corner to resize the region; both running totals re-converge to the new exact volume.