use integration to find the average value of A(t) = 3/t between values 2 and 8

To find the average value of the function \( A(t) = \frac{3}{t} \) between \( t = 2 \) and \( t = 8 \), we can use the formula for the average value of a function over an interval \([a, b]\):

\[
\text{Average value of } A(t) = \frac{1}{b - a} \int_{a}^{b} A(t) \, dt
\]

Here, \( A(t) = \frac{3}{t} \), \( a = 2 \), and \( b = 8 \). So, the formula becomes:

\[
\text{Average value} = \frac{1}{8 - 2} \int_{2}^{8} \frac{3}{t} \, dt
\]

### Step 1: Set up the integral

\[
\text{Average value} = \frac{1}{6} \int_{2}^{8} \frac{3}{t} \, dt
\]

### Step 2: Solve the integral

The integral of \( \frac{1}{t} \) is \( \ln|t| \). So, we can solve the integral as follows:

\[
\int_{2}^{8} \frac{3}{t} \, dt = 3 \int_{2}^{8} \frac{1}{t} \, dt = 3 [\ln|t|]_{2}^{8}
\]

Evaluating the definite integral:

\[
3 [\ln(8) - \ln(2)] = 3 \ln \left(\frac{8}{2}\right) = 3 \ln(4)
\]

Since \( \ln(4) = 2 \ln(2) \), we get:

\[
3 \ln(4) = 6 \ln(2)
\]

### Step 3: Calculate the average value

Now substitute the result of the integral back into the formula for the average value:

\[
\text{Average value} = \frac{1}{6} \times 6 \ln(2) = \ln(2)
\]

Thus, the average value of \( A(t) = \frac{3}{t} \) between \( t = 2 \) and \( t = 8 \) is:

\[
\boxed{\ln(2)}
\]

Would you like more details, or do you have any questions?

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Here are 5 related questions to explore further:
1. How do you find the average value of a quadratic function over an interval?
2. Can you calculate the average value of \( A(t) = \frac{5}{t^2} \) over the same interval?
3. What is the physical meaning of the average value of a function in an applied problem?
4. How does the result change if we compute the average of \( A(t) = \frac{3}{t} \) over a different interval?
5. What if we replace \( \frac{3}{t} \) with a trigonometric function like \( \sin(t) \)? How do we compute the average then?

**Tip:** When solving integrals involving \( \frac{1}{t} \), remember that the result is always related to the natural logarithm function \( \ln(t) \).

Learn how to calculate the average value of the function A(t) = 3/t between t = 2 and t = 8 using integration. This step-by-step guide covers key calculus concepts, including the formula for the average value of a function and the natural logarithm integral.

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