The Fundamental Theorem of Calculus

Two Faces Become One

Integration has worn two faces: antiderivatives (reversing differentiation) and definite integrals (accumulated area). The Fundamental Theorem of Calculus (FTC) unites them, proving accumulation and rate-of-change are inverse operations.

Theorem Part	Mathematical Rule	Core Meaning
Part 1	$F(x) = \int_a^x f(t)\, dt \implies F'(x) = f(x)$	Derivative of accumulated area is the curve's height
Part 2	$\int_a^b f(x)\, dx = F(b) - F(a)$	Practical evaluation rule via antiderivatives

Part 1 shows differentiation and integration undo each other. Part 2 is the practical powerhouse, eliminating tedious Riemann sums by using antiderivatives.

Two independently-built panels for the SAME curve, merging into one

Practical Power & Pitfalls

To compute $\int_0^2 x\, dx$ using Part 2, find the antiderivative $F(x) = \frac{x^2}{2}$ and evaluate at endpoints: $\left[\frac{x^2}{2}\right]_0^2 = 2 - 0 = 2$ , matching a triangle's area exactly.

Rule Aspect	Detail & Best Practice
Endpoint Order	Always upper minus lower: $F(b) - F(a)$
Sign Reversal	Reversing limits flips the final sign
Constant $+C$	Cancels out during subtraction; any antiderivative works

Common Pitfall: Subtracting endpoints in the wrong order, or attempting to use Part 2 on indefinite integrals. The $+C$ constant always vanishes in the subtraction.

Two Faces Become One

Practical Power & Pitfalls

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Calculus