Calculus 1

Calculus 1 introduces the fundamental concepts of limits, derivatives, and integrals, providing a foundation for advanced mathematical studies.

Understanding LimitsIntroduction to DerivativesDefinite and Indefinite Integrals
Continuity and DiscontinuityThe Chain Rule and Product RuleApplications of Derivatives
Calculus in PhysicsCalculus in Economics and BiologyCalculus in Technology and Engineering
Mastering Limit ProblemsDerivative and Integral Shortcuts
Understanding LimitsIntroduction to Deri...Definite and Indefin...Continuity and Disco...The Chain Rule and P...Applications of Deri...Calculus in PhysicsCalculus in Economic...Calculus in Technolo...Mastering Limit Prob...Derivative and Integ...
Basic Concepts

Definite and Indefinite Integrals

What Is an Integral?

Integrals are the opposite of derivatives. They help us add up infinitely many tiny pieces to find areas under curves, among other things. Imagine calculating the area of a wavy field—integrals do that!

Two Types of Integrals

  • Indefinite Integral: Represents a family of functions whose derivative is the original function.
  • Definite Integral: Calculates the net area under a curve between two points.

Notation

  • Indefinite: \( \int f(x) , dx \)
  • Definite: \( \int_{a}^{b} f(x) , dx \)

The Fundamental Theorem of Calculus

This connects derivatives and integrals: finding the area under a curve can be done by subtracting values of an antiderivative.

Why Are Integrals Important?

  • Calculate areas, volumes, and totals.
  • Model things like distance traveled or total growth.

Basic Integration Rules

  • Power Rule: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \) (for \( n eq -1 \))

Examples

  • The area under \( y = x \) from 0 to 2 is \( \int_0^2 x dx = 2 \).

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

In a Nutshell

Integrals help us add up tiny pieces to find areas and totals.

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Continuity and Discontinuity