Overview

This topic explores how definite integrals are used in real-world and geometric contexts — calculating area between curves, displacement from velocity, average values of functions, and accumulated change over time.

Key Concepts and Structures

• Area Between Curves: \int_a^b [f(x) - g(x)] dx, where f(x) is on top and g(x) is below
• Accumulated Change: The integral of a rate of change gives the total change. E.g., distance from velocity: \int v(t) dt
• Displacement vs. Total Distance:
  • Displacement: \int_a^b v(t) dt
  • Total distance: \int_a^b |v(t)| dt
• Average Value of a Function: f_{avg} = \frac{1}{b - a} \int_a^b f(x) dx
• Units: Integral of a rate (like m/sec) results in units of the base quantity (like meters)

Quick Tip

Practice Problems

1. Find the area between f(x) = x^2 and g(x) = x from x = 0 to x = 1
   Show Solution

   Top curve: g(x) = x; Bottom: f(x) = x^2

   \int_0^1 (x - x^2) dx = \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_0^1

   = (1/2 - 1/3) = 1/6

   Final Answer: 1/6

2. A particle moves with velocity v(t) = 3t^2 - 6. Find displacement from t = 0 to t = 3
   Show Solution

   \int_0^3 (3t^2 - 6) dt = \left[ t^3 - 6t \right]_0^3

   = (27 - 18) - (0 - 0) = 9

   Final Answer: 9 units