Question

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:

• A. \( 250 \)
• B. \( 210 \)
• C. \( 240 \)
• D. \( 220 \)

Hint:

To find the area between curves, set up an integral for the difference of the curves over the given interval. Make sure to simplify the integrand before solving the integral.

Answer 1

The Correct Option is A

Step 1: Determine Intersection Points

Equate the two equations to find intersection points:
\[ 4 - \frac{x^2}{4} = \frac{x - 4}{2} \]
Multiply by 4:
\[ 16 - x^2 = 2(x - 4) \]
\[ 16 - x^2 = 2x - 8 \]
Rearrange into a quadratic equation:
\[ x^2 + 2x - 24 = 0 \]
Solve the quadratic equation:
\[ x = \frac{-2 \pm \sqrt{4 + 96}}{2} = \frac{-2 \pm 10}{2} \]
The intersection x-values are \( x = 4 \) and \( x = -6 \).

Step 2: Identify Upper and Lower Functions

Test a point within the interval, e.g., \(x = 0\):
\[ \begin{align} \text{Parabola:} & \quad y = 4 - 0^2/4 = 4 \\ \text{Line:} & \quad y = (0 - 4)/2 = -2 \end{align} \]
The parabola \( y = 4 - \frac{x^2}{4} \) is the upper function over the interval \( [-6, 4] \).

Step 3: Formulate the Area Integral

The area \(\alpha\) is given by the integral of the upper function minus the lower function from \(x = -6\) to \(x = 4\):
\[ \alpha = \int_{-6}^{4} \left[\left(4 - \frac{x^2}{4}\right) - \left(\frac{x - 4}{2}\right)\right] dx \]
Simplify the integrand:
\[ \alpha = \int_{-6}^{4} \left(6 - \frac{x}{2} - \frac{x^2}{4}\right) dx \]

Step 4: Evaluate the Integral

Integrate term by term:
\[ \int_{-6}^{4} 6 \, dx = [6x]_{-6}^{4} = 6(4) - 6(-6) = 24 + 36 = 60 \\ \int_{-6}^{4} -\frac{x}{2} \, dx = \left[-\frac{x^2}{4}\right]_{-6}^{4} = -\frac{4^2}{4} - \left(-\frac{(-6)^2}{4}\right) = -4 - (-9) = 5 \\ \int_{-6}^{4} -\frac{x^2}{4} \, dx = \left[-\frac{x^3}{12}\right]_{-6}^{4} = -\frac{4^3}{12} - \left(-\frac{(-6)^3}{12}\right) = -\frac{64}{12} - \left(-\frac{-216}{12}\right) = -\frac{16}{3} - 18 = -\frac{16}{3} - \frac{54}{3} = -\frac{70}{3} \]
Combine the results:
\[ \alpha = 60 + 5 - \frac{70}{3} = 65 - \frac{70}{3} = \frac{195 - 70}{3} = \frac{125}{3} \]

Step 5: Calculate \(6\alpha\)

Multiply the area by 6:
\[ 6\alpha = 6 \times \frac{125}{3} = 2 \times 125 = 250 \]

Step 6: Select the Correct Option

The calculated value of \(6\alpha\) matches option (1).

Final Answer (1) 250