Fundamental Theorem of Calculus Part 1

Calculus 1 · Axiom Academy

LESSON The Fundamental Theorem, Part 1 Accumulated area is itself a function — and its rate of growth is the curve you started with. Start with a continuous function f(t) and a fixed left endpoint a . Now let the right endpoint be a variable, x , and record the area gathered so far. That record is a brand-new function. Feed it an x ; it returns the area from a out to x . 2. The Rate of Area Accumulation How fast does A grow? Push the right endpoint a little further, from x to , and ask how much area that bought you — per unit of x . That is the derivative, by definition. The extra area, divided by the extra width. The numerator is a thin sliver of area sitting between x and . Because f is continuous, over a short enough stretch the curve barely moves — so that sliver is very nearly a rectangle : width , height f(x) . Take the limit. As the "very nearly" becomes exact, the 's cancel, and what is left is startlingly simple. Width cancels; the height survives. If f is continuous on an interval containing a , then is differentiable and A'(x) = f(x) . The theorem is a licence to differentiate integrals on sight — even integrals nobody can evaluate in closed form. You do not need an antiderivative; you only need the integrand. Example 1 — straight from the theorem Let . No elementary antiderivative of exists — and it does not matter. The upper limit is plain x , so Part 1 applies directly: Example 2 — when the upper limit moves faster

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