The Definite Integral

Definition

The Definite Integral is the mathematical limit of a Riemann sum as the width of the partitions approaches zero. It represents the “net signed area” between a function and the $x$-axis over a closed interval $[a, b]$.

Why It Matters

The definite integral is the primary tool for calculating total accumulation in a continuous system. It allows us to determine exact quantities—such as total energy, mass, or distance—when the rate of change is constantly varying.

Core Concepts

• Limit of Sums: $\int_a^b f(x) dx = \lim_{\|P\| \to 0} \sum_{k=1}^n f(c_k) \Delta x_k$.
  • How to read: “The integral from a to b of f of x with respect to x equals the limit as the norm of the partition approaches zero, of the sum from k equals one to n of f of c k times delta x k.”
  • Meaning / when to use: Definite integral as limit of Riemann sums—rectangles of height $f(c_k)$ and width $\Delta x_k$ refine to exact signed area.
• Integrability: A function is “integrable” if this limit exists. Most common functions (continuous or with finite jumps) are integrable.
• Net Signed Area: Area above the $x$-axis is positive; area below is negative. The integral calculates the balance between the two.

Connected Concepts

• Properties of Definite Integrals
• The Natural Logarithm as an Integral
• Function Definition, Function Notation