MATHEMATICAL ANALYSIS 11
Fundamental Theorem of Calculus
The Fundamental Theorem explains why accumulation and differentiation are inverse operations.
Part I: differentiating accumulation
If \(f\) is continuous and
\[F(x)=\int_a^x f(t)\,dt,\]then
\[\boxed{F'(x)=f(x)}\]
Increasing \(x\) by a small amount adds a thin region whose area is approximately \(f(x)\Delta x\), so the accumulated area changes at rate \(f(x)\).
Part II: evaluating definite integrals
If \(F'=f\), then
\[\boxed{\int_a^b f(x)\,dx=F(b)-F(a)}\]
Why this is profound
The definite integral is defined through accumulation, while the derivative is defined through local limits. The theorem proves that these apparently different constructions are connected.
Conditions matter
The standard elementary statement assumes continuity. More advanced analysis extends integration beyond continuous functions, but the hypotheses must always be stated.