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To find the average value of a function \( f(r) \) on an interval \([a, b]\), we use the formula:

\[
\text{Average value} = \frac{1}{b - a} \int_a^b f(r) \, dr
\]

In this case, the function is given as:

\[
f(r) = \frac{4}{(1 + r)^2}
\]

and the interval is \([1, 6]\). Thus, we will compute:

\[
\text{Average value} = \frac{1}{6 - 1} \int_1^6 \frac{4}{(1 + r)^2} \, dr = \frac{1}{5} \int_1^6 \frac{4}{(1 + r)^2} \, dr
\]

Let's calculate this integral step by step.

### Step 1: Set up the integral

We need to solve:

\[
\int_1^6 \frac{4}{(1 + r)^2} \, dr
\]

This integral can be simplified using a substitution. Let \( u = 1 + r \), then \( du = dr \), and the limits of integration change from \( r = 1 \) to \( r = 6 \), corresponding to \( u = 2 \) and \( u = 7 \). The integral becomes:

\[
\int_2^7 \frac{4}{u^2} \, du
\]

### Step 2: Solve the integral

The integral of \( \frac{1}{u^2} \) is \( -\frac{1}{u} \). So, we compute:

\[
\int_2^7 \frac{4}{u^2} \, du = 4 \left[ -\frac{1}{u} \right]_2^7 = 4 \left( -\frac{1}{7} + \frac{1}{2} \right)
\]

Simplifying:

\[
= 4 \left( \frac{1}{2} - \frac{1}{7} \right) = 4 \times \frac{5}{14} = \frac{20}{14} = \frac{10}{7}
\]

### Step 3: Calculate the average value

Now, using the result of the integral, the average value is:

\[
\text{Average value} = \frac{1}{5} \times \frac{10}{7} = \frac{10}{35} = \frac{2}{7}
\]

Thus, the average value of the function on the interval \([1, 6]\) is \( \frac{2}{7} \).

Would you like more details or have any questions? Here are 5 related questions:

1. How does substitution work when integrating functions?
2. What is the geometric interpretation of the average value of a function?
3. Can the average value of a function be greater than the function's maximum value on the interval?
4. How can we generalize this method for other rational functions?
5. What happens to the average value if we change the limits of integration?

**Tip**: Always check the units of your answer if you're working with physical quantities, as integration often involves unit conversions.

Find the average value of the function f(r) = 4 / (1 + r)^2 on the interval [1,6].

Learn how to find the average value of the function f(r) = 4 / (1 + r)^2 over the interval [1,6]. This problem explores integral calculus, including the substitution method and application of the average value formula. Suitable for undergraduate or AP Calculus students.

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