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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is the bridge between the two halves of calculus. It reveals that differentiation and integration are inverse operations — and provides the practical tool for evaluating definite integrals without computing limits of Riemann sums.

In this lesson

  1. Why this theorem is extraordinary
  2. Part 1: Derivatives of integral functions
  3. Part 2: The evaluation theorem
  4. Applying the theorem
  5. Historical and mathematical significance

1 Why This Theorem Is Extraordinary

Before the Fundamental Theorem, derivatives and integrals were studied as separate problems. Derivatives came from tangent line problems; integrals came from area problems. They seemed to have nothing to do with each other.

Newton and Leibniz, independently in the 17th century, discovered that these apparently unrelated operations are inverses of each other — as closely related as addition and subtraction, or multiplication and division. This unification created modern calculus and made it practical to compute integrals without laboriously summing rectangles.

2 Part 1: Derivatives of Integral Functions

If F(x) = ∫ₐˣ f(t)dt, then F'(x) = f(x).

In words: if you define a function by integrating f from a fixed point a to a variable upper limit x, then the derivative of that function is just f(x) itself. Integration and differentiation undo each other.

Intuition for Part 1

F(x) is the accumulated area under f from a to x. As x increases by a tiny amount dx, the area increases by approximately f(x)·dx — a thin sliver of height f(x) and width dx. So the rate of change of F with respect to x is f(x). That's F'(x) = f(x).

3 Part 2: The Evaluation Theorem

If F is any antiderivative of f (meaning F'(x) = f(x)), then: ∫ₐᵇ f(x)dx = F(b) − F(a)

This is the practical part. To compute a definite integral, find any antiderivative F, evaluate it at the endpoints, and subtract. No limits of sums needed.

Using the Fundamental Theorem
Compute ∫₁⁴ (2x + 1) dx
1Find an antiderivative: F(x) = x² + x (since F'(x) = 2x + 1)
2Evaluate at upper limit: F(4) = 16 + 4 = 20
3Evaluate at lower limit: F(1) = 1 + 1 = 2
4Subtract: F(4) − F(1) = 20 − 2 = 18
Answer: ∫₁⁴ (2x+1)dx = 18
Area Under a Parabola
Compute ∫₀² x² dx
1Antiderivative: F(x) = x³/3
2F(2) = 8/3, F(0) = 0
3∫₀² x² dx = 8/3 − 0 = 8/3 ≈ 2.667
Answer: The area under y=x² from 0 to 2 is 8/3

4 Applying the Theorem

The theorem makes definite integrals computable. Steps: (1) find an antiderivative F(x), (2) evaluate F(b) − F(a). The notation F(x)|ₐᵇ means "evaluate F from a to b."

Don't forget the constant of integration cancels

When computing ∫ₐᵇ f(x)dx = F(b)−F(a), the +C cancels: (F(b)+C)−(F(a)+C) = F(b)−F(a). So you can ignore +C for definite integrals.

5 Historical and Mathematical Significance

The Fundamental Theorem is one of the most important results in all of mathematics. It unified two previously separate fields, created a practical calculus that could be applied to physics and engineering, and launched the scientific revolution of the 17th and 18th centuries.

Without it, computing areas, volumes, arc lengths, and probabilities would require laborious numerical approximation for every problem. With it, these computations often reduce to simple algebraic evaluation. It's why calculus-based physics, engineering, and economics are even possible at their current level of sophistication.

Practice Problems

Compute ∫₀³ x² dx
Antiderivative: x³/3. F(3)−F(0) = 27/3 − 0 = 9
Compute ∫₁⁴ (3x + 2) dx
F(x) = 3x²/2 + 2x. F(4) = 24+8=32. F(1) = 1.5+2=3.5. 32−3.5 = 28.5
If F(x) = ∫₀ˣ t² dt, what is F'(x)?
By Part 1 of the FTC: F'(x) = x²