(1) Basic Integration Rules & Problems, Riemann Sum, Area, Sigma Notation, Fundamental Theorem, Calculus

Integration Basics

  • Focus on basic integration rules

  • Cover methods for finding areas under curves:

    • Riemann sums

    • Left endpoints

    • Right endpoints

    • Midpoint rule

    • Trapezoidal rule

  • Difference between definite and indefinite integrals

  • Properties of definite integrals

  • Area calculation using geometric figures or shapes

  • Fundamental theorem of calculus

  • Particle motion problems

Antiderivatives and Integration

  • Finding Antiderivatives

    • Example: Anti-derivative of (x^2 , dx)

      • Differentiate: (d(x^3)/dx = 3x^2)

      • Anti-derivative: ( \int x^2 dx = \frac{1}{3}x^3 + C)

    • Process:

      1. Add one to the exponent

      2. Divide by new exponent

Examples of Antiderivatives

  • ( \int x^4 , dx = \frac{1}{5}x^5 + C)

  • ( \int 4x^7 , dx = \frac{4}{8}x^8 + C = \frac{1}{2}x^8 + C)

  • ( \int 13x^8 , dx = \frac{1}{9}x^9 + C)

  • ( \int 3x , dx = \frac{3}{2}x^2 + C)

  • Constant example: ( \int 5 , dx = 5x + C)

Multiple Terms

  • Example: ( \int (4x - 7) , dx = 2x^2 - 7x + C)

Radical Functions

  • Example: ( \int \sqrt{x} , dx = \int x^{1/2} , dx = \frac{2}{3}x^{3/2} + C)

Exponential Functions

  • Example: ( \int e^x , dx = e^x + C)

  • Special cases when integrating

Riemann Sums

  • Area under curves through approximation

  • Left and right endpoint sums

    • Left: sum using left side of rectangles

    • Right: sum using right side of rectangles

  • Midpoint Rule: Average of left and right endpoints.

  • Trapezoidal Rule: Use average height of sections for approximation.

Definite vs. Indefinite Integrals

  • Definite Integrals: Evaluate from A to B, resulting in a numerical value.

  • Indefinite Integrals: Results in a function plus C (constant of integration).

  • Fundamental theorem of calculus relates the two.