SEC2_GYM · Calculus · CALC.INTEGRAL.CONCEPT

Integral as accumulation and area

A definite integral measures signed accumulation and can be understood as net area.

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Intuition

Understand first, compress later.

Split the region into many thin rectangles. As their width tends to zero, their area sum approaches the integral.

Switch representations

Riemann sumf(x) = x, n = 8

curve fexact area = 88 rectangles (right edge) = 9

Right-endpoint rule: each of the 8 rectangles is as tall as the curve at the right edge of its strip (width Δx = 0.5). That leaves a visible seam between rectangle top and curve: R₈ = 9 against the exact 8 — 1 too much. More strips shrink that seam to nothing — that passage to the limit is the integral.

Formal view

is the limit of suitable Riemann sums. The fundamental theorem connects derivatives and integrals.

Why does it work?

Integration adds infinitesimal contributions and is therefore the natural counterpart to the local rate of change measured by derivatives.