Step 1: Understanding the Question:
The problem involves dividing an area under a parabola into two equal parts using a vertical line $x = \alpha$. Step 2: Key Formula or Approach:
Area under $y = f(x)$ from $a$ to $b$ is $\int_a^b f(x) dx$. Step 3: Detailed Explanation:
The curve is $y = \frac{x^2}{4}$.
Total area $A$ from $x=0$ to $x=4$:
\[ A = \int_0^4 \frac{x^2}{4} dx = [\frac{x^3}{12}]_0^4 = \frac{64}{12} = \frac{16}{3} \]
The line $x = \alpha$ divides it into equal areas, so the area from $0$ to $\alpha$ is $A/2$:
\[ \int_0^\alpha \frac{x^2}{4} dx = \frac{1}{2} \times \frac{16}{3} = \frac{8}{3} \]
\[ [\frac{x^3}{12}]_0^\alpha = \frac{8}{3} \]
\[ \frac{\alpha^3}{12} = \frac{8}{3} \]
\[ \alpha^3 = \frac{8 \times 12}{3} = 32 \]
\[ \alpha = (32)^{\frac{1}{3}} \] Step 4: Final Answer:
The value of $\alpha$ is $(32)^{\frac{1}{3}}$.