Integral calculus

Definition

Integral calculus is the study of accumulation and the areas under curves. It focuses on the integral, which represents the total value of a quantity that is changing.

Why It Matters

Without integration, we can only see snapshots. Integration allows us to understand the cumulative impact of change—how speed becomes distance, how flow becomes volume, and how marginal gains become total wealth.

Core Concepts

• The Definite Integral: Formally $\int_{a}^{b} f(x) \, dx$, representing the net area between the function $f(x)$ and the x-axis from $a$ to $b$.

  • How to read: “The integral from a to b of f of x with respect to x.”
  • Meaning / when to use: Sums infinitely many thin vertical slices of height $f(x)$ and width $dx$; the result is net signed area—positive where $f > 0$, negative where $f < 0$.

• Accumulation: If $v(t)$ is speed, the integral $\int v(t) \, dt$ represents the total distance traveled.

  • How to read: “The integral of v of t with respect to t.”
  • Meaning: Distance is the running total of speed over time; the indefinite integral recovers position up to a constant of integration.

• Antiderivatives: The process of finding a function whose derivative is the original function.

Connected Concepts

• Calculus
• The Definite Integral
• Fundamental Theorem of calculus
• Limits