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The image shows a problem that asks to find a function \( f(x) \) such that the given integral represents the arc length of the function over the interval \([0, 2]\). The integral is:

\[
\int_0^2 \sqrt{1 + (8x)^2} \, dx
\]

We are asked to find the function \( f(x) \) such that this integral corresponds to the arc length of \( f(x) \). The formula for the arc length of a function \( f(x) \) from \( x = a \) to \( x = b \) is:

\[
L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx
\]

In this case, comparing the given integral to the arc length formula, we can observe that:

\[
\sqrt{1 + (f'(x))^2} = \sqrt{1 + (8x)^2}
\]

This implies:

\[
(f'(x))^2 = (8x)^2
\]

Taking the square root of both sides:

\[
f'(x) = 8x
\]

To find \( f(x) \), integrate both sides:

\[
f(x) = \int 8x \, dx = 4x^2 + C
\]

Since no initial condition is given, we can set \( C = 0 \), so the function is:

\[
f(x) = 4x^2
\]

Thus, the function \( f(x) = 4x^2 \) is the one for which the integral represents the arc length.

Would you like further explanation, or do you have any questions?

Here are 5 related questions for deeper understanding:
1. What is the geometric interpretation of arc length?
2. How would you derive the arc length formula from first principles?
3. How does the arc length formula change if \( f(x) \) is parameterized differently?
4. What would the integral look like for a different function, say \( f(x) = x^3 \)?
5. Can arc length be computed for functions that are not smooth?

**Tip:** When computing arc lengths, always ensure that the derivative \( f'(x) \) is continuous and differentiable, as this ensures the function is smooth and the arc length formula applies directly.

Name a function f(x) for which the following integral represents the arc length of the function on the interval [0, 4]:
∫(from 0 to 2) √(1 + (8x)^2) dx

This problem involves finding the function f(x) for which the given integral represents the arc length on the interval [0, 2]. It covers concepts such as integral calculus, arc length, and differentiation, along with key formulas for solving arc length problems.

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