AB Unit 8: Applications of Integration

Applications of Integration

8.1 Finding the Average Value of a Function on an Interval

• Concept Overview: The average value of a function over an interval [a, b] is calculated based on the area under the curve divided by the width of the interval.
  • If f(x) is a constant function, then it is represented as:
    $f(x) = k$
  • The area of the rectangle is given by: $R = k(b - a)$
    • Therefore, the average value formula is:
      ext{Average Value} = rac{1}{b - a} ext{´} \int_{a}^{b} f(x) dx

Example 1: Average value of f(x) over [0, 3], where f(x) = x² + 2

• To find the average value: ext{Average} = rac{1}{3 - 0} \int_{0}^{3} (x^2 + 2) \,dx
  • Solve the integral:
    = rac{1}{3} \left[ rac{x^3}{3} + 2x
    ight]_{0}^{3} (After evaluating at bounds this would yield the average value)

Example 2: Average value of f(x) over [π/2, π], where f(x) = 2cos(x)

• Similar procedure will be applied with integrating from π/2 to π:
  • First set up the integral:
    ext{Average} = rac{1}{rac{ au}{2} - rac{ au}{2}} \int_{rac{ au}{2}}^{ au} 2 \cos(x) \, dx

Practice Problems

• Problem 1: Find the average value of $y = x^2 \sqrt{x^3 + 1}$ over the interval [0, 2].
• Problem 2: A variable rate of water draining modeled by: $r(t) = \begin{cases} 600t, & 0 \leq t \leq 5 \ 1000e^{-0.21(t - 5)}, & t > 5 \end{cases}$
  • Find the average drainage rate between t = 0 and t = 8 hours.

Connecting Position, Velocity, and Acceleration Functions

• Definitions:
  • Position (s(t)): The location of a particle over time
  • Velocity (v(t)): The rate of change of position: $v(t) = s'(t)$
  • Acceleration (a(t)): The rate of change of velocity: $a(t) = v'(t)$
• Displacement Calculation:
  If the acceleration is given by $a(t) = 4t$, determine the final position with initial values.

Examples related to motion

• Velocity and Position:
  • If $v(t)$ is defined by integration of $a(t)$ and initial conditions such as $v(0) = -2$, integrate to find the position at specific times.
  • Example: If the velocity model yields positions like:
  • Total Distance: $\int_{a}^{b} |v(t)| \,dt$

Interpretation of Results

• Average height of a function is represented as the area divided by the interval width, emphasizing the fundamental relationship in calculus linking area under a curve and average value.