Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus explained — both parts, with examples showing how differentiation and integration are inverse operations.
The Formula — Part 1
The Fundamental Theorem of Calculus (FTC) is the central result connecting differentiation and integration — two operations that appear completely different but are in fact inverses of each other. It has two parts, each serving a distinct purpose.
Part 2 — The Evaluation Theorem
Where F is any antiderivative of f (i.e. F'(x) = f(x)). This is the part used in everyday calculus — it turns the problem of computing a definite integral into simple evaluation at two points.
Variables
| Symbol | Meaning | Note |
|---|---|---|
| f(x) | The integrand (function being integrated) | Must be continuous on [a, b] |
| F(x) | Antiderivative of f(x) | F'(x) = f(x) |
| a, b | Lower and upper limits of integration | a ≤ b |
| ∫ab | Definite integral from a to b | Returns a number |
Example 1 — Polynomial
Evaluate ∫13 x² dx
Step 1: Find antiderivative F(x) = x³/3
Step 2: Evaluate F(3) − F(1) = 27/3 − 1/3 = 9 − 0.333
= 26/3 ≈ 8.667
Example 2 — Trigonometric
Evaluate ∫0π sin(x) dx
Step 1: Antiderivative of sin(x) is −cos(x)
Step 2: [−cos(π)] − [−cos(0)] = −(−1) − (−1) = 1 + 1
= 2
Example 3 — Part 1 in Action
If g(x) = ∫0x √(1 + t³) dt, find g'(x)
By FTC Part 1, differentiation undoes integration directly
g'(x) = √(1 + x³)
Why It Matters
Before the FTC (proven independently by Newton and Leibniz around 1670), computing areas under curves required exhausting geometric limit arguments for every new problem. The FTC made integration mechanical — find any antiderivative, plug in the limits, subtract. This single result is the foundation of all applied calculus.
When to Use It
- Computing areas, volumes, arc lengths, and surface areas
- Finding displacement from a velocity function
- Calculating work done by a variable force
- Probability — computing probabilities from density functions
- Any problem involving accumulation over a continuous interval