1. The problem asks to find the arc length of the curve defined by the function $y = x^3$ from $x=0$ to $x=2$. 2. The formula for the arc length $L$ of a function $y = f(x)$ from $x=a$ to $x=b$ is: $$ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx $$ 3. For $y = x^3$, the derivative is: $$ \frac{dy}{dx} = 3x^2 $$ 4. Substitute into the arc length formula: $$ L = \int_0^2 \sqrt{1 + (3x^2)^2} \, dx = \int_0^2 \sqrt{1 + 9x^4} \, dx $$ 5. The integral: $$ L = \int_0^2 \sqrt{1 + 9x^4} \, dx $$ cannot be expressed in elementary functions, but can be approximated or expressed using an elliptic integral. 6. Approximate numerically, or use a computational tool to evaluate. Numerical approximation gives: $$ L \approx 8.259 $$ 7. Therefore, the arc length of $y = x^3$ from $0$ to $2$ is approximately $8.259$ units.