If the area of the region bounded by the curves $y = 4 - \frac{x^2}{4}$ and $y = \frac{x - 4}{2}$ is equal to $\alpha$, then $6\alpha$ equals:

Step 1: Find Intersection Points

To find the points where the two curves intersect, we set their $y$ -values equal to each other. This will give us the $x$ -coordinates of the intersection points, which define the limits of integration for calculating the area.

Step 2: Solve for x

We simplify the equation by multiplying by 4 to clear the denominators and rearrange it into a standard quadratic form. Factoring the quadratic equation $x^2 + 2x - 24 = 0$ yields the roots $x = -6$ and $x = 4$. These are the lower and upper limits of integration, respectively.

Step 3: Determine Upper and Lower Curves

We need to identify which curve is the upper curve and which is the lower curve within the interval $[-6, 4]$. By testing a point, for example $x=0$, we find $y_1(0) = 4$ and $y_2(0) = -2$. Since $4 > -2$, the parabola $y_1$ is the upper curve and the line $y_2$ is the lower curve.

Step 4: Set up the Integral for Area

The area $\alpha$ between two curves is found by integrating the difference between the upper curve and the lower curve over the interval of intersection. We set up the definite integral from $x = -6$ to $x = 4$ with the parabola as the upper function and the line as the lower function.

Step 5: Evaluate the Integral

We evaluate the definite integral by finding the antiderivative of the integrand and then applying the Fundamental Theorem of Calculus. We substitute the upper limit, then the lower limit, and subtract the results to find the area $\alpha$.

Step 6: Calculate 6α

Finally, we multiply the calculated area $\alpha$ by 6 to get the required value.

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