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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.

Find the average value of the function f(x) = x^2 on the interval [3, 4].

Learn how to calculate the average value of the function f(x) = x^2 over the interval [3, 4]. This solution uses calculus concepts such as integration and the fundamental theorem of calculus, providing a detailed step-by-step process. Suitable for students in Grades 11-12 and early college.

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