To find the average value of a function $f(x) = x^2$ over the interval $[3, 4]$, we use the formula for the average value of a function:

$\text{Average Value} = \frac{1}{b - a} \int_a^b f(x) \, dx$

In this case:

• $f(x) = x^2$
• $a = 3$, $b = 4$

Now, let's compute the average value:

$\text{Average Value} = \frac{1}{4 - 3} \int_3^4 x^2 \, dx = \int_3^4 x^2 \, dx$

I will now compute the integral.The average value of the function $f(x) = x^2$ on the interval $[3, 4]$ is approximately $12.33$.

The exact value of the integral is $\frac{37}{3}$, and the corresponding average value is $\frac{37}{3} \approx 12.33$.

Would you like more details on this solution or have any other questions?

Here are some related questions:

1. How do we interpret the average value of a function geometrically?
2. What happens to the average value when the interval is extended?
3. How would the result change if the function were $x^3$ instead of $x^2$?
4. Can this formula be applied to any continuous function?
5. How does this concept apply in real-world scenarios?

Tip: The average value of a function over an interval gives insight into the "central" or "typical" value the function takes on that interval.