Qwen: Qwen3 VL 235B A22B Thinking passed on Does AI know AP Calculus AB?: "Mathematically correct"

To evaluate the definite integral:

$$
\int_{0}^{4} 3x^2 \, dx
$$

we begin by finding the **antiderivative** of the integrand $ 3x^2 $. 

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### Step 1: Find the Antiderivative

The power rule for integration states:

$$
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(for } n \neq -1\text{)}
$$

Applying this to $ 3x^2 $:

$$
\int 3x^2 \, dx = 3 \cdot \int x^2 \, dx = 3 \cdot \left( \frac{x^3}{3} \right) = x^3 + C
$$

So, the antiderivative is $ x^3 $ (the constant $ C $ disappears in a definite integral).

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### Step 2: Apply the Limits of Integration

Now, evaluate the antiderivative at the upper and lower bounds:

$$
\left[ x^3 \right]_0^4 = 4^3 - 0^3 = 64 - 0 = 64
$$

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### Final Answer

$$
\boxed{64}
$$