Fundamental Theorem of calculus, Part 2

Definition

The Fundamental Theorem of calculus (FTC) Part 2, also known as the Evaluation Theorem, provides the practical tool for calculating definite integrals. It states that the net area under a curve can be found by evaluating its antiderivative at the endpoints.

Why It Matters

FTC Part 2 is the workhorse of the modern world; it transforms the impossible task of summing infinite tiny slices into a simple subtraction problem, enabling the efficient design of everything from bridge spans to neural networks.

Core Concepts

• The Formula: $\int_a^b f(x) dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$.
  • How to read: “The integral from a to b of f of x with respect to x equals F of b minus F of a.”
  • Meaning / when to use: Find net accumulation by evaluating any antiderivative at the endpoints — no Riemann sums needed. This is the main tool for computing definite integrals.
• Evaluation Notation: Often written as $[F(x)]_a^b$ or $F(x) \Big|_a^b$.
  • How to read: “The function F of x evaluated from a to b, or F of x with a vertical bar from a to b.”
  • Meaning: Shorthand for $F(b) - F(a)$ — plug in the top limit, subtract the bottom limit.
• Efficiency: This eliminates the need to compute complex limits of Riemann sums, making integration as accessible as basic subtraction.

Connected Concepts

• Fundamental Theorem of calculus, Part 1
• Fundamental Theorem of Line Integrals
• Graphs of Functions: Properties