Applications of Integration

8. Applications of Integration

8.1 Finding the Average Value of a Function on an Interval

  • Average Value Definition: The average value of a function over an interval [a, b] can be visualized as the height of a rectangle whose area equals the area under the curve of the function.

  • Mean Value Theorem for Integrals:

    • Theorem 8.1.1: If f(x) is continuous on [a, b], then there exists a number c in the interval
      [a, b] such that:

    [ \int_a^b f(x) \, dx = f(c) (b - a) ]

    • Significance: This theorem guarantees the existence of c, but does not provide a method to find it directly.
8.1.2 Average Value of a Function

  • The average value of f on [a,b] is given by:

    [ f{\text{avg}} = \frac{1}{b - a} \inta^b f(x) \, dx ]

  • Important Distinction: Average value is not the same as the average rate of change.

    • Average Velocity: The average rate of change of position over time is defined as:

    [ v_{\text{avg}} = \frac{s(b) - s(a)}{b - a} ]

    • Using the integral, average velocity can also be expressed as:

    [ v{\text{avg}} = \frac{1}{b - a} \inta^b v(t) \, dt ]

Example 8.1: Finding the Average Value of a Function

  • Function: f(x) = 3x² - 2x

  • Interval: [1, 4]

  • Calculation of average value follows:

    [ f{\text{avg}} = \frac{1}{4 - 1} \int1^4 (3x^2 - 2x) \, dx ]

  • Solve for definite integral:

    [ \int_1^4 (3x^2 - 2x) ] results in an average value of 16.

8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
  • The fundamental relationships using derivatives are expressed as:
    • [ \frac{ds}{dt} = v(t) ] (Position to Velocity)
    • [ \frac{dv}{dt} = a(t) ] (Velocity to Acceleration)
Example 8.2: Velocity of a Particle

  • Given v(t) = 9t²:

    • Displacement on [1, 4] is calculated using:

    [ \int_1^4 v(t) \, dt ]

    • Total Distance: Must consider the absolute value of velocity, leading to:

    [ \int_a^b |v(t)| \, dt ]

Exercise 8.2: Average velocity and analysis of motion
  • Given a(t) = 2t - 6, find:
    (a) Average velocity over [0, 6]
    (b) Conditions for moving left (v(t) < 0)
    (c) Total distance traveled can utilize integrals.
8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
Example 8.3: Chemical Flow Rate

  • Flow rate function: R(t) = (180+3t) liters/minute.

  • Calculate amount of chemical entering in the first 20 minutes using:

    [ \int_0^{20} (180 + 3t) \, dt ]

Exercise 8.4: Water Leakage
  • Model: F(t) = 5 + t - 4 sin(t²)
  • Find total gallons from Tuesday (day 2) to Thursday (day 4) using definite integrals.
8.4 Finding the Area Between Curves Expressed as Functions of x
Theorem 8.4.1: Area Between Two Curves

  • For functions f and g continuous over [a, b]:

    [ A = \int_a^b (f(x) - g(x)) \, dx ]

Example 8.4: Area Calculation
  • Enclosed region by y = x² + 2, y = x, and x = 0 to x = 1 can be set up as a definite integral.
Example 8.5: Additional Area Calculation
  • Area between y = x² + 1 and y = 5 involves setup of appropriate limits and functions.
8.5 Finding the Area Between Curves Expressed as Functions of y
Theorem 8.5.1: Area of Region in y-Orientation

  • Continuous functions f and g over [a, b]:

    [ A = \int_a^b (f(y) - g(y)) \, dy ]

Example 8.6: Area Calculation with y-ordering
  • Bounded by x = 3y² and x = y + 1.
  • Set up and execute integral according to intersections.
8.6 Finding the Area Between Curves That Intersect at More Than Two Points
Example 8.7: Area Between Polynomial Functions
  • Calculate area between f(x) = 3x³ + x² + 10x and g(x) = x² + 2x, ensuring to find intersection points to address all bounded areas in calculations.