Calculus I · Unit 3A · lesson

Fundamental Theorem of Calculus, Part I

Concept

Learning objectives

Differentiate accumulation functions and explain why the derivative recovers the integrand.

Fundamental Theorem of Calculus, Part I

Explanation

Why the derivative of accumulated area returns the integrand

Suppose $A(x)=\int_a^x f(t)\,dt$ . Increasing $x$ by a small amount $h$ adds a thin strip whose width is $h$ and whose height is close to $f(x)$ . The added accumulation is therefore approximately $f(x)h$ , so the quotient $[A(x+h)-A(x)]/h$ is approximately $f(x)$ . In the limit, the approximation becomes exact under the theorem's hypotheses.

This result is profound because it says accumulation has a local rate, and that rate is precisely the function being accumulated. It is the mathematical statement that a running total changes at the current input rate. The theorem turns area and accumulation functions into differentiable objects and explains why integration and differentiation are inverse processes rather than unrelated units placed next to each other by tradition.

Theorem

FTC Part I

If $f$ is continuous and

A(x)=\int_a^x f(t)\,dt,

then

A'(x)=f(x).

Why? Increasing $x$ by a small amount $h$ adds a thin strip whose width is $h$ and whose height is approximately $f(x)$ . Thus

\frac{A(x+h)-A(x)}h =\frac1h\int_x^{x+h}f(t)\,dt \approx f(x).

The approximation becomes exact in the limit.

Guided walkthrough

Differentiate without evaluating the integral

G(x)=\int_2^x\sqrt{1+t^4}\,dt,

then

\boxed{G'(x)=\sqrt{1+x^4}}.

No elementary antiderivative is needed.

Interactive checku3a-ftc1-01

If $F(x)=\int_0^x\cos(t^2)\,dt$ , find $F'(x)$ .

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Show hint

FTC Part I returns the integrand evaluated at the upper bound.

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Optional advanced note

Continuity is sufficient, not the final word

FTC Part I is commonly stated for continuous integrands because that hypothesis is easy to check and strong enough for a clean theorem. More advanced integration theories weaken the hypotheses and clarify exactly where the derivative of an accumulation function exists.

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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