Average Value of a Curve

The average value of a function, particularly over a defined interval \( [a, b] \), offers valuable insights in various fields. It provides a representative mean level of the function’s behavior, useful in physics for average energy or velocity calculations, in economics for average trends, and in engineering for signal processing and material properties analysis. This concept is fundamental in understanding and simplifying complex functional behaviors over specific ranges.

Finding the average value of a function over an interval \( [a, b] \) involves a few steps. Here’s how to do it:

1. Identify the Function and Interval:

• Choose the function \( f(x) \) whose average value you need to find.
• Define the interval \([a, b]\) over which the average value is to be calculated.

2. Set Up the Integral:

• The formula for the average value of a function \( f(x) \) over the interval \([a, b]\) is given by:
  \( \frac{1}{b – a} \int_{a}^{b} f(x) \, dx \)
• This integral represents the total area under the curve of \( f(x) \) from \( a \) to \( b \).

3. Compute the Integral:

• Calculate the definite integral \(\int_{a}^{b} f(x) \, dx\). This can be done using various methods, such as anti-derivatives, numerical integration, or computational tools, depending on the complexity of the function.

4. Divide by the Length of the Interval:

• After finding the integral, divide the result by the length of the interval, which is \( b – a \). This step essentially averages the total area under the curve over the width of the interval.

5. Interpret the Result:

• The result gives you the average value of the function \( f(x) \) over the interval \([a, b]\). It represents the mean height of the function on this interval, providing a single value that summarizes the behavior of the function over that range.

In summary, the average value of a function over an interval is found by integrating the function over that interval and then dividing by the interval’s length. This process yields a meaningful summary of the function’s behavior across the specified range.

Let’s find the average value of the function \( f(x) = e^x \) over the interval \([1, 3]\).

1. Compute the Integral:

• The integral of \( e^x \) from \( 1 \) to \( 3 \) is:
  \( \int_{1}^{3} e^x \, dx \)
• The antiderivative of \( e^x \) is \( e^x \) itself. So, we evaluate \( e^x \) from \( 1 \) to \( 3 \).

2. Calculate and Interpret:

• Substitute the limits into the antiderivative and calculate the difference:
  \( e^3 – e^1 = e^3 – e \)
• Then divide by the length of the interval, which is \( 3 – 1 = 2 \):
  \( \frac{1}{2} \times (e^3 – e) \)

The average value of the function \( f(x) = e^x \) over the interval \([1, 3]\) is about \( 8.684 \). This value represents the mean level of the function’s behavior across this specific range.