Question

If the area bounded by the curve \(x^2=4 y, \mathrm{X}\)-axis and the line \(x=4\) is divided into equal areas by the line \(x=\alpha\), then the value of \(\alpha\) is …

• A. \(\frac{1}{32}\)
• B. 32
• C. \((32)^{\frac{1}{2}}\)
• D. \((32)^{\frac{1}{3}}\)

Hint:

Compute the full area from x = 0 to 4, then find alpha so that the area from 0 to alpha is half of it.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The curve is \(y=\frac{x^2}{4}\). The area between the curve, the X-axis and \(x=4\) is found by integration.

Step 2: Total area
\(A=\int_0^4\frac{x^2}{4}dx=\frac{1}{4}\cdot\frac{64}{3}=\frac{16}{3}\).

Step 3: Half the area
\(\int_0^{\alpha}\frac{x^2}{4}dx=\frac{\alpha^3}{12}=\frac{8}{3}\), so \(\alpha^3=32\).

Step 4: Result
\(\alpha=(32)^{\frac{1}{3}}\), option (D).

Final Answer:
The value of alpha is \((32)^{1/3}\). \[ \boxed{\text{(D) }(32)^{\frac{1}{3}}} \]