A double integral adds signed height times a tiny patch of area across a region.
The displayed columns use midpoint heights over a rectangular partition. For the bowl the exact integral is 4ab(a² + b²)/3; for the saddle it is 4ab(a² − b²)/3. The wave sin x cos y integrates to zero on this symmetric region by odd symmetry in x.
A signed integral is not the same as total geometric volume between the graph and the plane.
Drag to rotate, or use the view sliders. The surface and level curve are sampled; all point values and derivatives use exact formulas.