AP CALCULUS · MODULE 03A

Definite & Indefinite Integrals
The Antiderivative

The integral is the reverse of differentiation — finding area under curves, accumulating quantities, and the Fundamental Theorem of Calculus.

📘 BUders Learning System

This module covers antiderivatives, Riemann sums, definite integrals, and the Fundamental Theorem of Calculus.

What is an Antiderivative?

Definition: $F'(x) = f(x)$, indefinite integral notation $\int f(x) dx = F(x) + C$.

definition → start

Basic Integration Rules

Power rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, constant multiple, sum/difference, $\int e^x dx$, $\int \frac{1}{x} dx$.

rules → start

Left, right, midpoint, and trapezoidal sums — approximating area under a curve.

approximation → start

Definite Integrals

Definition $\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$, properties, geometric interpretation.

definition → start

Fundamental Theorem of Calculus

FTC Part 1: $\frac{d}{dx} \int_a^x f(t) dt = f(x)$, FTC Part 2: $\int_a^b f(x) dx = F(b) - F(a)$.

theorem → start

Practice Problems

20 solved problems covering all topics in this module.

practice → start

20-question interactive quiz with instant feedback.

assessment → start