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For the arc length formula $\int_{a}^{b} \sqrt{1 + (f'(x))^2} dx$ to be valid, the function $f(x)$ must be continuous on the closed interval $[a, b]$, and its derivative $f'(x)$ must also be continuous on $[a, b]$. These conditions ensure that the curve is smooth enough for the integral to accurately represent its length.

Conditions: The curve is defined by a function $y = f(x)$.; The interval of integration is $[a, b]$.; $f(x)$ is continuous on $[a, b]$.; $f'(x)$ is continuous on $[a, b]$.

For the arc length formula $\int_{a}^{b} \sqrt{1 + (f'(x))^2} dx$ to be valid, the function $f(x)$ must be continuous on the closed interval $[a, b]$, and its derivative $f'(x)$ must also be continuous on $[a, b]$. These conditions ensure that the curve is smooth enough for the integral to accurately represent its length.

Conditions: The curve is defined by a function $y = f(x)$.; The interval of integration is $[a, b]$.; $f(x)$ is continuous on $[a, b]$.; $f'(x)$ is continuous on $[a, b]$.

The final exact value obtained for the displayed arc-length integral is $\frac{208}{27}$. This is calculated by multiplying the outside constant $\frac{8}{27}$ by the evaluated bracket difference 26.

Conditions: The integral has been fully substituted and evaluated up to the step $\frac{8}{27}\cdot 26$.; The arithmetic simplification $8 \cdot 26 = 208$ is performed exactly.

The final exact value obtained for the displayed arc-length integral is $\frac{208}{27}$. This is calculated by multiplying the outside constant $\frac{8}{27}$ by the evaluated bracket difference 26.

Conditions: The integral has been fully substituted and evaluated up to the step $\frac{8}{27}\cdot 26$.; The arithmetic simplification $8 \cdot 26 = 208$ is performed exactly.