Calculus Lecture Notes: Introduction to Chapter 4

Overview

• Chapter 4 Transition: Shift from derivatives to solving the area under a curve.
• Main Problems in Calculus:
  • Previously: Finding the slope of a curve at a point (derivatives).
  • Now: Finding the area under a curve.

The Area Under a Curve

• Concept: For a continuous function from A to B, determine the area between the curve and the x-axis.
• Methods:
  • Rectangular Method: Use rectangles to approximate the area.
  • Anti-Derivative Method: A more concise analytical approach.

Rectangular Method

• Process:
  1. Divide the interval into n equal-width sub-intervals.
  2. Create rectangles within each sub-interval.
  3. Determine rectangle height using left, right, or midpoint endpoints.
  4. Sum the areas of rectangles to approximate the area under the curve.
• Improvement: Increase the number of rectangles to improve approximation accuracy.

Anti-Derivative Method

• Concept: If the derivative of an area function equals the original function, find the anti-derivative to determine the area under the curve.
• Key Insight: The derivative of the area function gives back the original function.
• Application: Use differentiation rules in reverse to find anti-derivatives.

Integration and Anti-Derivatives

• Anti-Derivative Definition: A function F(x) is an anti-derivative of f(x) if its derivative equals f(x).
• Integral Notation:
  • ∫ f(x) dx = F(x) + C, where C is the constant of integration.

Basic Integration Rules

• Integral of constants, powers, and basic trigonometric functions.
• Use known derivatives to find corresponding integrals (e.g., ∫ cos(x) dx = sin(x) + C).

Properties of Integrals

1. Constants can be factored out of integrals.
2. Integrals of sums and differences can be split.
3. Integrals do not distribute over multiplication.

Solving Differential Equations

• Initial Value Problems: Given a differential equation and an initial condition, find a specific solution.
• Example Approach:
  1. Integrate the given derivative function.
  2. Apply initial conditions to solve for constants.

Application: Projectile Motion

• Problem Setup: Calculate the trajectory of a projectile launched vertically.
• Key Equations:
  • Position, velocity, and acceleration functions.
  • Use initial conditions to determine specific function details.
• Process:
  1. Derive position and velocity functions using integration.
  2. Use maximum velocity condition (velocity = 0) to find peak height.
  3. Solve position equation for when the projectile hits the ground.

Conclusion

This lecture transitions calculus students from understanding derivatives to integrating functions to find areas under curves, focusing on real-world applications like projectile motion. The introduction of anti-derivatives and integration lays the foundation for solving a wide array of mathematical and physical problems.