The function f  is given by f(x) = 3x^2 + 1 . What is the average value of  f over the closed interval [1,3]?

To find the average value of a function \( f(x) \) over a closed interval \([a, b]\), we use the formula:

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

Given \( f(x) = 3x^2 + 1 \) and the interval \([1, 3]\), we can apply this formula.

### Step 1: Set up the integral
Here, \( a = 1 \) and \( b = 3 \). So, the formula becomes:

\[
\text{Average value} = \frac{1}{3-1} \int_1^3 (3x^2 + 1) \, dx
\]

\[
\text{Average value} = \frac{1}{2} \int_1^3 (3x^2 + 1) \, dx
\]

### Step 2: Evaluate the integral

First, compute the integral:

\[
\int_1^3 (3x^2 + 1) \, dx = \int_1^3 3x^2 \, dx + \int_1^3 1 \, dx
\]

1. Evaluate \( \int_1^3 3x^2 \, dx \):

   \[
   \int 3x^2 \, dx = x^3 \quad \text{(using the power rule)}
   \]

   So,

   \[
   \int_1^3 3x^2 \, dx = \left[ x^3 \right]_1^3 = 3^3 - 1^3 = 27 - 1 = 26
   \]

2. Evaluate \( \int_1^3 1 \, dx \):

   \[
   \int_1^3 1 \, dx = x \Big|_1^3 = 3 - 1 = 2
   \]

Combine the results:

\[
\int_1^3 (3x^2 + 1) \, dx = 26 + 2 = 28
\]

### Step 3: Calculate the average value

Now plug the integral result back into the average value formula:

\[
\text{Average value} = \frac{1}{2} \times 28 = 14
\]

Thus, the average value of \( f \) over the interval \([1, 3]\) is \(\boxed{14}\).

Would you like further clarification or have any more questions?

### Related Questions:
1. How do you find the average value of \( g(x) = 4x - 2 \) over the interval \([-2, 2]\)?
2. What is the average value of the function \( h(x) = \sin(x) \) over the interval \([0, \pi]\)?
3. How can you interpret the average value of a function geometrically?
4. What is the average value of \( k(x) = e^x \) over the interval \([0, 1]\)?
5. How does changing the interval \([a, b]\) affect the average value of a function?

### Tip:
When finding the average value of a function over an interval, make sure to integrate correctly and carefully handle any algebraic steps. Use symmetry or known results to simplify the integration process where possible.

Discover how to find the average value of the function f(x) = 3x^2 + 1 over the interval [1, 3]. Step-by-step solution included using integration techniques. Suitable for advanced high school students.

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