AP Calculus AB

Advanced Placement Calculus AB covering limits, derivatives, and integrals.

Understanding LimitsIntroduction to DerivativesFundamentals of Integrals
Techniques of DifferentiationApplications of DerivativesDefinite Integrals and the Fundamental Theorem
Calculus in Physics and EngineeringCalculus in Economics and BiologyCalculus and Everyday Problems
Mastering Multiple-Choice QuestionsShowing Work and Earning Free Response Points
Understanding LimitsIntroduction to Deri...Fundamentals of Inte...Techniques of Differ...Applications of Deri...Definite Integrals a...Calculus in Physics ...Calculus in Economic...Calculus and Everyda...Mastering Multiple-C...Showing Work and Ear...
Basic Concepts

Fundamentals of Integrals

What Is an Integral?

Integrals are the opposite of derivatives. They help you find the total accumulation, like the area under a curve or the total distance traveled.

Types of Integrals

  • Definite Integrals: Calculate the area between the curve and the x-axis over an interval.
  • Indefinite Integrals: Find a general formula for accumulated change.

Notation

  • \( \int f(x) , dx \) is the indefinite integral of \( f(x) \).
  • \( \int_{a}^{b} f(x) , dx \) is the definite integral from \( a \) to \( b \).

How Do We Calculate Integrals?

  • Reverse the Power Rule: Add 1 to the exponent, then divide by the new exponent.
  • Use Substitution: For tricky functions, change variables.

Why Are Integrals Important?

Integrals help us solve problems involving total quantities—like distance, area, and even probability!

Examples

  • The area under \( y = x \) from \( x = 0 \) to \( x = 2 \) is 2.

  • The indefinite integral of \( f(x) = 3x^2 \) is \( x^3 + C \).

In a Nutshell

Integrals let us add up small pieces to find totals, such as area or accumulated change.

Key Terms

Integral
A mathematical tool for finding the accumulated sum or area under a curve.
Antiderivative
A function whose derivative is the original function.
Definite Integral
The integral of a function over a specific interval, giving a numeric value.
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Techniques of Differentiation