Mean Value Theorem for Integrals and Applications

The Mean Value Theorem for Integrals

  - Definition: If a function $f(x)$ is continuous on the closed interval $[a, b]$, then there exists at least one number $c$ in the interval such that the following holds:
     f(c) = rac{1}{b-a} imes ext{Area under } f(x) ext{ from } a ext{ to } b

• Visual Demonstration:
    - Example given where the area under $f(x)$ from $x=0$ to $x=4$ calculated to be approximately $21.3$.
    - The implication is that there exists a $c$ such that
     f(c) = rac{1}{4 - 0} imes 21.3 ext{ or } f(c) = rac{21.3}{4} ext{ which gives us the average value of } f(x).

Finding the Average Value of a Function on an Interval

• Learning Objective: CHA-4.B: Determine the average value of a function using definite integrals.

Average Value Calculation

• Formula: If $f(x)$ is integrable on a closed interval $[a, b]$, the average value of $f(x)$ is given by:
     ext{Average Value} = rac{1}{b-a} imes extstyleigint_{a}^{b} f(x) ext{ dx}
  

Examples

Example 1: Using Actual Function Values

• Let ( f(x) = 2\cdotrac{(x^2)}{12} ) over the interval $[2, 5]$.
• Step A: Find the average value of $f(x)$ on the given interval.
• Step B: Find $c$ such that the average value of $f(x)$ equals $f(c)$.

Example 2: Integrating Function

• Given function is $f(x) = x^2 - 3$ over the interval $[1, 4]$.
• Step A: Calculate average value.
• Step B: Determine value(s) of $c$ per the Mean Value Theorem for Integrals.

Application of Integration – Estimation of Milk Volume

Example 3: Got Milk?

• Scenario: Cows are milked over a period of 2 hours. Model milk volume in the tank as a function $M(t)$, where $M(t)$ is increasing and $M(0) = 300$ gallons.
• Table Data: Given values for $M(t)$ at specific times:
      - $M(0) = 300$
      - $M(20) = 780$
      - $M(50) = 1640$
      - $M(90) = 3360$
      - $M(120) = 4250$
  
• Tasks:
    - a. Estimate total volume pumped into the tank over the 2-hour span; explain units.
    - b. Use table data for evaluation.
    - c. Approximate with a right Riemann sum using 4 subintervals from the table, explaining if it overestimates or underestimates.

Particle Motion – Integration Applications

Concept Overview

• Recall the motion equations for a particle moving on a line: Position $s(t)$, velocity $v(t)$, and acceleration $a(t)$.

Example 1: Using Technology

• Given velocity $v(t)$ and the position $s(t)$.
    - Find times when speed is 5, integral expression for position.
    - Analyze the direction changes and speed increases/decreases.

Expressions and Their Meanings in Motion

Volume Calculations Using Known Cross Sections

Typical Shapes and Volume Formulas

• Square: Side length=s, Area= $s^2$, Volume= $ext{Area} imes ext{Height}$.
• Rectangle: Area= $lw$, Volume= $A imes h$.
• Semi-Circle: Radius $r$, Area= $rac{1}{2} imes ext{area of full circle}$, Volume= Area $ imes$ Height.
• Triangular Cross Sections: Area= rac{1}{2} ase imes height.

Volume of Solids of Revolution

• Disk Method:
    - For rotation around x-axis:
    V = extstyle igint_{a}^{b} ext{Area}(R(x)) ext{ dx} = extstyle igint_{a}^{b} rac{ ext{Area}(R^2)}{ ext{height}} ext{ dx}
• Washer Method:
    - Volume expressed as:
    V = extstyle igint_{a}^{b} ig( R(y)^2 - r(y)^2 ig) dy

Applications of Definite Integrals in Accumulation Problems

  - Interpreting integrals to model physical phenomena like snow accumulation or entries in a contest.
  - Careful analysis leads to dealing with continuous functions in practical applications.