Lecture Notes: Introduction to Definite Integrals

Transition from Indefinite to Definite Integrals

• Indefinite Integrals: Involves an unknown constant (+ C); represents an area function without a specific area.
• Definite Integrals: Transition allows calculation of specific area under a curve or between two points.
• Area Calculation: Similar to derivatives calculated with limits, integrals also use limits.

Concepts of Riemann Sums

• Riemann Sum: Method to approximate the total area under a curve by summing areas of multiple rectangles.
• Arbitrary Points: Rectangles can have left, right, or midpoints, but the specific point doesn’t matter.
• Limits and Integrals: As the number of rectangles increases, their width approaches zero, forming an integral.

Understanding the Integral Notation

• Integral Symbol (∫): Represents adding an infinite number of rectangles under a curve from point A to B.
• dx: Represents the infinitesimally small width of rectangles.
• Example: Integral from A to B of a function f(x) represents the total area from A to B.

Geometric Perspective of Integrals

• Example: Integral of 2 from 1 to 4

  • Function is a horizontal line at y=2.
  • Area Calculation: Base (3) x Height (2) = Total Area of 6 square units.

• Example: Integral of (x+2) from 1 to 2

  • Graph x+2: Line starting at y=2 with slope 1.
  • Area Calculation: Divide into rectangle and triangle, calculate individually, add together.

• Example: Integral of √(1-x²) from 0 to 1

  • Represents the upper half of a unit circle (radius = 1).
  • Area Calculation: Quarter circle area = π/4.

Properties of Definite Integrals

1. Zero Area: When upper and lower limits are equal.
2. Negative Integral: Reversing limits of integration negates the integral.
3. Constant Multiplication: Constants can be factored out of integrals.
4. Addition/Subtraction: Integrals can be split by addition/subtraction of functions.
5. Splitting Integrals: Integrals can be split over an interval [A, B] into [A, C] and [C, B].
6. Positive/Negative Areas:
   • If f(x) >= 0 for all x in [A, B], integral is >= 0.
   • If f(x) <= 0 for all x in [A, B], integral is <= 0.

Practical Example

• Integral from 0 to 1 of (4 - 2√(1-x²)) dx
  • Split into separate integrals.
  • Use geometric methods to find areas.
  • Combine results to find total area.

Conclusion

• Theoretical understanding of definite integrals involves both geometric interpretation and algebraic manipulation.
• Next steps include computational techniques for solving integrals more directly.