If the area of the region \[ \{(x,y) : 0 \leq y \leq \min\{2x, 6x - x^2\}\} \] is \( A \), then \( 12A \) is equal to \ldots
Correct Answer: 304
Solution and Explanation
We have:
\(A = \frac{1}{2} \int_{4}^{6} x \cdot (6x - x^2) \, dx.\)
Calculating the integral:
\(A = \frac{1}{2} \int_{4}^{6} (6x - x^2) \, dx = \frac{76}{3}.\)
Multiplying by 12:
\(12A = 12 \times \frac{76}{3} = 304.\)
The Correct answer is: 304
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