Question

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

Answer 1

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