Fundamental Theorem of Calculus: Parts, Examples & Proof

There’s a moment in every calculus class when the lightbulb goes on: the sudden understanding that differentiation and integration aren’t separate skills but two sides of the same coin. That connection is the fundamental theorem of calculus (FTC), a result that ties the two branches together and makes modern mathematics work.

Discoverers: Isaac Newton and Gottfried Wilhelm Leibniz ·
Stated: Late 17th century ·
Part 1: Derivative of an integral ·
Part 2: Definite integral via antiderivative ·
Also Called: Newton-Leibniz theorem

Five key facts, one pattern: the FTC transforms the problem of summing tiny pieces into a simple difference of antiderivatives.

What is the fundamental theorem of calculus?

Understanding the connection between derivatives and integrals

The fundamental theorem of calculus states that differentiation and integration are inverse operations (GeeksforGeeks (educational resource)). It consists of two parts: Part 1 gives a way to differentiate an integral, and Part 2 lets you evaluate a definite integral using an antiderivative.

This insight means that the two main operations of calculus are really the same process in reverse. Without the FTC, computing areas under curves would require summing infinitely many rectangles every time. With it, you just find a function whose derivative matches the integrand and evaluate at the endpoints.

The implication: once you master derivatives, integrals become far more manageable.

What is FTC 1 and FTC 2?

Part 1: The derivative of an integral

FTC Part 1 (often called the First Fundamental Theorem) says: if f is continuous on [a,b] and F(x) = ∫_a^x f(t) dt, then F'(x) = f(x) for all x in (a,b) (Cuemath (learning platform)). In plain terms, taking the derivative of an integral returns the original function.

This result is what lets you find the derivative of a function defined as an integral. For example, d/dx ∫_{1.5}^x (t² + 3t) dt = x² + 3x (GeeksforGeeks (educational resource)).

Part 2: Evaluating definite integrals using antiderivatives

FTC Part 2 (the Second Fundamental Theorem) provides the method we actually use: if f is continuous on [a,b] and F is any antiderivative of f, then ∫_a^b f(x) dx = F(b) – F(a) (OpenStax Calculus (open-access textbook)). This turns area calculations into simple arithmetic.

The catch: you must remember that the antiderivative F need not be unique; any two antiderivatives differ by a constant. That constant disappears when you subtract F(b) – F(a).

How is the fundamental theorem of calculus proved?

Outline of the proof for Part 1

The proof of Part 1 uses the definition of derivative as a limit and applies the Mean Value Theorem for integrals. For F(x) = ∫_a^x f(t) dt, consider F'(x) = lim_{h→0} [F(x+h) – F(x)]/h = lim_{h→0} (1/h) ∫_x^{x+h} f(t) dt. By continuity, the average value of f on [x, x+h] approaches f(x) as h→0 (Cuemath (learning platform)).

Outline of the proof for Part 2

Part 2 follows from Part 1. Define g(x) = ∫_a^x f(t) dt. By Part 1, g'(x)=f(x), so g is an antiderivative. For any antiderivative F, F(x) = g(x) + C. Then ∫_a^b f(x) dx = g(b) – g(a) = F(b) – F(a) (Lumen Learning (university courseware)).

The pattern: the proof elegantly shows that the derivative of the accumulation function recovers the original integrand, and that any antiderivative can be used for evaluation.

What are examples of the fundamental theorem of calculus?

Example: Computing ∫₀¹ x² dx

Using Part 2: find antiderivative F(x) = x³/3. Then ∫₀¹ x² dx = F(1) – F(0) = 1³/3 – 0³/3 = 1/3 (GeeksforGeeks (educational resource)).

Example: Using FTC to find derivative of an integral

By Part 1, d/dx ∫₀ˣ sin(t) dt = sin(x). When the upper limit is a function of x, say d/dx ∫₃ˣ² (3+t)/(1+t³) dt, you apply the chain rule together with FTC Part 1 (Cuemath (learning platform)).

What is the fundamental theorem of calculus used for?

Applications in physics: displacement, velocity, acceleration

In physics, the FTC links position, velocity, and acceleration. If s(t) is position, then velocity v(t) = s'(t), and acceleration a(t) = v'(t). The definite integral of velocity over time gives net displacement – exactly Part 2 in action (OpenStax Calculus (open-access textbook)).

Applications in economics: consumer and producer surplus

Economists use FTC to compute total benefit from marginal benefit curves. The area under a marginal benefit curve is total benefit, obtained by integrating the marginal function (Lumen Learning (university courseware)).

What this means: the FTC is not a theoretical curiosity – it’s the tool that lets any field quantify accumulated change.

The fundamental theorems of calculus relate derivatives and integrals with one another.

– Wolfram MathWorld (reference resource)

The FTC is extremely powerful for relating differentiation and integration.

– Lumen Learning Calculus 2 (university courseware)

For students grappling with calculus, the choice is clear: invest time in understanding both parts of the FTC, or risk tackling integrals with a blindfold. The theorem isn’t just a formula – it’s the conceptual bridge that makes the rest of calculus possible.

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