Calculus I · Unit 3A · lesson

Fundamental Theorem of Calculus, Part II

Concept

Learning objectives

Evaluate definite integrals with antiderivatives and connect the result to net accumulated change.

Fundamental Theorem of Calculus, Part II

Explanation

Why antiderivatives evaluate exact accumulation

Riemann sums define the definite integral, but evaluating a limit of sums directly would be unbearable for most functions. The second part of the Fundamental Theorem provides the computational shortcut: if $F'=f$ , then $\int_a^b f(x)\,dx=F(b)-F(a)$ . The endpoint difference captures all of the tiny accumulated contributions at once.

The notation $[F(x)]_a^b$ should be read as an instruction to substitute both endpoints and subtract in the correct order. The constant of integration cancels, which is why definite integrals do not need $+C$ . Even so, the theorem is not merely a formula to memorize. It depends on recognizing an antiderivative and on understanding that the result is net accumulation, with sign and units inherited from the original integrand and variable.

Theorem

FTC Part II

If $f$ is continuous on $[a,b]$ and $F'(x)=f(x)$ , then

\int_a^bf(x)\,dx=F(b)-F(a).

The theorem is astonishing because it turns a limit of many sums into two endpoint evaluations.

Guided walkthrough

Evaluate a definite integral

\begin{aligned} \int_1^3(2x+4)\,dx &=[x^2+4x]_1^3\\ &=(9+12)-(1+4)\\ &=16. \end{aligned}

The bracket notation means evaluate the antiderivative at the upper endpoint and subtract its value at the lower endpoint.

Interactive checku3a-ftc2-01

Evaluate $\int_0^3 2x\,dx$ .

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Show hint

An antiderivative is $x^2$ .

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Common mistake

The definite integral is not $F(b)+F(a)$ , and the lower endpoint is not optional. Write the bracket step explicitly until endpoint subtraction is automatic.

After the explanation

Use the section idea

Reading lens

Use the Fundamental Theorem as the bridge between accumulation functions, local rates, and endpoint evaluation.

Mental model

Differentiating a running total recovers its current integrand, while evaluating an accumulated total subtracts antiderivative endpoint values.

Decision

Separate FTC Part I, FTC Part II, net change, and variable-bound chain-rule tasks before manipulating notation.

Common trap

Forgetting a chain-rule factor at a variable bound, reversing endpoint subtraction, or adding +C to a definite value.

Check yourself

Can you state which part of the theorem applies and why its hypotheses and bounds fit?

Source & rights

Original instruction with traceable references.

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