The Main Idea 

• Definition of Definite Integral:
  • Generalizes the concept of area under a curve
  • [latex]\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x[/latex]
  • Function [latex]f(x)[/latex] is integrable if this limit exists
• Components of Definite Integral Notation:
  • [latex]\int[/latex]: Integration symbol (elongated [latex]S[/latex])
  • [latex]a, b[/latex]: Limits of integration (lower and upper)
  • [latex]f(x)[/latex]: Integrand
  • [latex]dx[/latex]: Variable of integration
• Integrability:
  • Continuous functions on [latex][a,b][/latex] are integrable
  • Some discontinuous functions may also be integrable
• Evaluation Methods:
  • Using the definition (Riemann sums)
  • Geometric formulas for area
  • More advanced techniques (to be learned later)

The Main Idea 

• Net Signed Area:
  • Represents the area between the curve and the [latex]x[/latex]-axis, taking into account the sign of the function
  • [latex]\int_a^b f(x) dx = A_1 - A_2[/latex]
  • [latex]A_1[/latex]: Area above [latex]x[/latex]-axis, [latex]A_2[/latex]: Area below [latex]x[/latex]-axis
  • Can be positive, negative, or zero
• Total Area:
  • Represents the total area between the curve and the [latex]x[/latex]-axis, regardless of the function’s sign
  • [latex]\int_a^b |f(x)| dx = A_1 + A_2[/latex]
  • Always non-negative
• Interpretation of Definite Integrals:
  • For positive functions: Area under the curve
  • For functions that change sign: Net signed area
  • Using absolute value: Total area
• Application to Displacement:
  • Velocity function [latex]v(t)[/latex]: Area under curve represents displacement
  • Net signed area: Final position relative to starting point
  • Total area: Total distance traveled

The Main Idea 

• Basic Properties of Definite Integrals:
  • Zero interval:
    • [latex]\int_a^a f(x) dx = 0[/latex]
  • Reversing limits:
    • [latex]\int_b^a f(x) dx = -\int_a^b f(x) dx[/latex]
  • Sum rule:
    • [latex]\int_a^b [f(x)+g(x)] dx = \int_a^b f(x) dx + \int_a^b g(x) dx[/latex] 
  • Difference rule:
    • [latex]\int_a^b [f(x)-g(x)] dx = \int_a^b f(x) dx - \int_a^b g(x) dx[/latex] 
  • Constant multiple:
    • [latex]\int_a^b cf(x) dx = c \int_a^b f(x) dx[/latex]
  • Splitting interval:
    • [latex]\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx[/latex]
• Comparison Theorem (for [latex]a \le b[/latex]): 
  • Non-negative function:
    • If [latex]f(x) \ge 0[/latex] for [latex]a \le x \le b[/latex], then [latex]\int_a^b f(x) dx \ge 0[/latex] 
  • Comparing functions:
    • If [latex]f(x) \ge g(x)[/latex] for [latex]a \le x \le b[/latex], then [latex]\int_a^b f(x) dx \ge \int_a^b g(x) dx[/latex]
  • Bounded function:
    • If [latex]m \le f(x) \le M[/latex] for [latex]a \le x \le b[/latex], then [latex]m(b-a) \le \int_a^b f(x) dx \le M(b-a)[/latex]

The Main Idea 

• Definition of Average Value:
  • For a function [latex]f(x)[/latex] continuous on [latex][a,b][/latex], the average value is: [latex]f_{\text{ave}} = \frac{1}{b-a} \int_a^b f(x) dx[/latex]
• Interpretation:
  • Generalizes the concept of arithmetic mean to continuous functions
  • Represents the height of a rectangle with base [latex][a,b][/latex] and equal area to that under the curve of [latex]f(x)[/latex]
• Derivation:
  • Based on Riemann sums:
    • [latex]\frac{1}{b-a}\lim_{n\to \infty}\sum_{i=1}^{n} f(x_i^*) \Delta x[/latex]
  • Limit of Riemann sum becomes the definite integral
• Applications:
  • Physics: Average velocity, average power, etc.
  • Economics: Average cost, average revenue, etc.
  • Statistics: Expected value of a continuous random variable