Volume Under a Surface

Calculus 3 · Axiom Academy

A curved roof over a flat floor traps a solid of air. Measure it by paving the floor with tiny boxes — then shrink them until the answer is exact. Measuring the space trapped under a curved roof In single-variable calculus you found the area under a curve by slicing a region into thin rectangles and adding them up. Move to a surface z = f(x,y) sitting above a flat region R in the plane, and the same trick measures the volume of the solid caught between the surface and the floor — you just pave R with little boxes instead of slicing with rectangles. Watch the region get paved with columns that rise to touch the surface. A coarse grid is a lumpy guess; as the boxes get smaller the stack fills the solid snugly, and the running box-total climbs to the exact volume. That total is the double integral . The volume under the surface is just accumulated box-volume — the running total the shrinking columns leave behind. Shrink the boxes yourself and watch the gap close Slide the handle to chop R into a finer and finer grid. Each column samples the surface's height at the middle of its little tile and contributes to the total. Watch the box-sum tighten onto the true volume — the gap between your estimate and the exact answer vanishing as the tiles shrink. As the box-sum becomes the double integral: What one box is worth: height times its footprint

This is the written version of the interactive lesson above. See the full Calculus 3 course.