Key Concept: If F'(x) = f(x) and f is continuous on [a,b], then integral(a to b) f(x) dx = F(b) - F(a).

Part 1 (FTC-1): If G(x) = integral(a to x) f(t) dt, then G'(x) = f(x). The integral is an antiderivative.

Part 2 (FTC-2): integral(a to b) f(x) dx = F(b) - F(a) for any antiderivative F.

Cue Questions:

• Why does the constant C cancel? Because F(b) + C - F(a) - C = F(b) - F(a).
• Can we always apply FTC? Only when f is continuous on [a,b]. If f has discontinuities, split the integral.

Summary: FTC connects the two central problems of calculus (area and tangent). It says computing areas reduces to finding antiderivatives.