1.4 Integration Formulas and the Net Change Theorem

Learning Objectives

• Apply basic integration formulas.
• Explain the net change theorem.
• Use the net change theorem for applied problems.
• Apply integrals of odd and even functions.

Overview

• Indefinite vs. Definite Integrals:
  • Definite Integral: A number or function, depending on variable limits.
  • Indefinite Integral: Represents a family of functions differing by a constant.
  • Choose the correct integral based on the problem.

Basic Integration Formulas

• Use previously studied integration formulas for application.
• Example: Using the power rule for integration.
  • Calculate definite integrals over an interval.

The Net Change Theorem

• New value = Initial value + Integral of rate of change.
• Important for applications in area, distance, volume, etc.
• Example: Car motion, calculating net displacement.
  • North at 40 mph, south at 30 mph.
  • Net displacement:
    • Calculate using definite integrals of velocities over time intervals.

Application Examples

• Finding Net Displacement:
  • Use velocity functions to calculate net displacement.
• Total Distance Traveled:
  • Integrate absolute values of velocity functions.

Applying the Net Change Theorem

• Example: Gasoline consumption in a motorboat.
  • Calculate how much gasoline is used over time.
• Example: Iceboat speed, distance calculation based on wind speed function.

Integrating Even and Odd Functions

• Even Functions:
  • Symmetric about y-axis.
  • Integral over [-a, a] equals twice the integral from [0, a].
• Odd Functions:
  • Symmetric about origin.
  • Integral over [-a, a] equals zero.

Example Problems

• Integrating Even/Odd Functions:
  • Verify integration formulas for even and odd functions.
• Exercise Examples:
  • Solve integration problems using basic formulas.
  • Calculate changes in perimeter, area, volume of geometric shapes.

Application in Physics and Real-Life Scenarios

• Calculate displacement and total distance using velocity functions.
• Use integration for fluid dynamics, mechanical scenarios, etc.
• Economic scenarios like energy consumption, tax calculations, etc.

Exercises

• Practice using integration formulas for various functions.
• Apply integration to real-world problems.

This section provides key integration techniques and their applications, especially focusing on the net change theorem, and covers the integration of even and odd functions. The exercises reinforce understanding through practical application scenarios.