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.