Volume by Cross-Sectional Area

Calculus 1 · Axiom Academy

LESSON Volume by Cross-Sectional Area Slice a solid into paper-thin slabs, add their volumes, and let the slices shrink to nothing — the sum becomes an integral. Lay the solid along the x -axis from x=a to x=b . At each position x , cut perpendicular to the axis. The flat face you expose is the cross-section , and its area depends on where you cut — call it A(x) . Give that slab a small thickness , and it's nearly a prism: its volume is the face area times the thickness. The cross-section at x has area A(x) A slab of thickness holds about of volume One slab is an estimate; the whole solid is the sum of all of them . Chop [a,b] into n equal pieces of width , take the cross-section at each sample point x_i , and add up the slab volumes. With only a few slabs the staircase is rough — but watch what happens as n climbs. Thick slabs over- or under-shoot the true shape — the estimate is crude. Area of its face times its width: . Thin slabs wrap the shape tightly — the gaps shrink toward zero. Each slab's error comes from the cross-section changing across its width. Halve the width and you roughly halve that change per slab — so the total error keeps falling as the slabs get thinner. Now push . The width , the staircase melts onto the true surface, and the Riemann sum becomes an integral . That limit is the exact volume — an infinite sum of infinitely thin slabs, written in one clean line.

This is the written version of the interactive lesson above. See the full Calculus 1 course.