6.4 FTC & Accumulation Functions

Accumulation function

An accumulation function has the form g(x) = ∫[a to x] f(t) dt — it integrates from a constant a up to a variable upper limit x.

FTC Part 1

The Fundamental Theorem of Calculus (Part 1) says g'(x) = f(x): differentiating an integral recovers the integrand. Differentiation and integration are inverse processes. With a chain rule: d/dx ∫[a to x²] f(t) dt = f(x²)·2x.

Key terms

Accumulation function

g(x) = ∫[a to x] f(t) dt.

FTC Part 1

g'(x) = f(x).

Inverse processes

Differentiation undoes integration.

With chain rule

d/dx ∫[a to x²] f(t) dt = f(x²)·2x.

Meaning

Differentiating an integral recovers the integrand.

Try an AP-style question

If g(x) = ∫₁ˣ t² dt, what is g'(x)?

1. x³/3
2. x²
3. 2x
4. x² − 1