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.