AP Calculus AB
Advanced Placement Calculus AB covering limits, derivatives, and integrals.
Basic Concepts
Introduction to Integrals
What Is an Integral?
Integrals are all about accumulation—adding up tiny pieces to find the whole. If derivatives break things down, integrals build them up!
The Idea
An integral calculates the total area under a curve between two points. This can represent distance, total amount, or accumulated change.
Notation
The integral of \( f(x) \) from \( a \) to \( b \) is written as \( \int_a^b f(x),dx \).
Why Use Integrals?
- Find areas under curves.
- Calculate total distance from velocity.
- Determine accumulated growth, like population or money over time.
Basic Types
- Definite Integrals: Give a specific value (area).
- Indefinite Integrals: Find a general formula (antiderivatives).
Examples
The area under \( y = x \) from 0 to 2 can be found using an integral: \( \int_0^2 x,dx = 2 \).
To find how far you've traveled if you know your speed at every moment, integrate your speed over time.
In a Nutshell
Integrals find the total or the area under curves.