6.4 The Fundamental Theorem of Calculus and Accumulation Functions

The Fundamental Theorem of Calculus (Part 2)

If f is continuous on [a, b], then ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f. This connects derivatives and integrals: integration undoes differentiation. It gives an exact way to evaluate definite integrals without using limits of Riemann sums.

Accumulation functions

An accumulation function g(x) = ∫ₐˣ f(t) dt represents the net accumulated area from a to x. It builds a new function of x from f. Values of g can be found using areas, given data, or the FTC itself — accumulation functions model total change over an interval.

Differentiating accumulation functions (FTC Part 1)

If g(x) = ∫ₐˣ f(t) dt, then g′(x) = f(x) — the derivative of an accumulation function returns the original integrand. With the chain rule, if the upper limit is u(x), g′(x) = f(u(x))·u′(x). This links accumulation directly to rates of change.

Key terms

Fundamental Theorem of Calculus (Part 2)

∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of continuous f.

Accumulation function

g(x) = ∫ₐˣ f(t) dt; represents the net accumulated area under f from a to x.

FTC Part 1

If g(x) = ∫ₐˣ f(t) dt, then g′(x) = f(x).

FTC with chain rule

If g(x) = ∫ₐ^(u(x)) f(t) dt, then g′(x) = f(u(x))·u′(x).

Why FTC matters

It links differentiation and integration, allowing definite integrals to be evaluated exactly using antiderivatives instead of limits of sums.

Try an AP-style question

If g(x) = ∫₁ˣ t² dt, what is g'(x)?

1. 2x
2. x² − 1
3. x²
4. x³/3