Chapter Overview

Main Focus

• Switch from derivatives to finding the area under a curve.
• Two main problems in calculus:
  1. Finding the slope of a curve at a point (done through derivatives).
  2. Finding the area under a curve (focus of chapters four and five).

Methods for Finding Area Under a Curve

Rectangular Method

• Concept: Divide the interval into equal sub-sections, create a rectangle in each sub-interval.
• Steps:
  1. Split interval into n equal parts.
  2. Use left endpoints, right endpoints, or midpoints to determine the height of each rectangle.
  3. Find the area of each rectangle and sum them.
  4. As the number of rectangles (n) approaches infinity, the approximation becomes exact.

Anti-Derivative Method

• Concept: Undo the derivative to find the original function representing the area.
• Steps:
  1. Pretend the given function is a derivative.
  2. Find the anti-derivative which gives the area function.

Practical Use

• Use geometry to verify the concept:
  • Example: Function f(x) = x + 1, find the area from 1 to x using triangle area formula.
  • Confirm the area using anti-derivative method.

Indefinite Integral & Properties

Integral Notation and Basic Integration

• Definition: The integral of f(x) with respect to x gives the anti-derivative F(x) + C.
• Properties:
  • Integral of a constant times a function:
  • Sum and difference rule:

Basic Integrals to Remember

• Polynomial Function:
• Key Trigonometric Integrals:
  • 
  • 
  • 
  • 

Differential Equations and Initial Value Problems

• Basic Concept: Given a derivative, find the original function.
• Example: If dy/dx = x^4, then find y such that dy/dx = x^4
  • Solution involves finding the integral and adding C.
• Initial Value Problems: Provides a specific value for C by giving an initial condition like y(0) = 1 to solve for C.

Application: Catapult Problem

• Given: A projectile is launched from a height of 16 ft with an initial velocity of 128 ft/sec.
• Goals:
  • Find the position function s(t).
  • Determine the maximum height.
  • Find when the projectile hits the ground.
• Using Physics: Integrate acceleration -32 ft/s^2 to find velocity and then position.
• Include initial conditions to determine constant C.

Steps to Solve Catapult Problem

1. Integrate acceleration to find velocity and solve for C using initial velocity.
2. Integrate velocity to find position and solve for C using initial height.
3. Set position function equate to zero to find when the projectile hits the ground.
4. Set velocity function to zero to find the time of maximum height
5. Substitute this time into the position function to find the maximum height.

Note: Ensure proper units and plug values back into equations for verification.