Question

If the area bounded by the parabola $x^2=4y$, the x-axis and the line $x=4$ is divided into two equal area by the line $x=\alpha$, then the value of a is:

• A. $2^{1/3}$
• B. $2^{2/3}$
• C. $2^{4/3}$
• D. $2^{5/3}$

Answer 1

Answer: (D)

Given : $x^2=4y\Rightarrow y=\frac{1}{4}{x}^2$
$\text{Area}={\int }_0^{4}ydx$
$\Rightarrow \text{Area}=\frac{1}{4}{\int }_0^4{x}^{2}dx=\frac{1}{4}{\left[\frac{x^3}{3}\right]}_0^4=$
$\frac{1}{12}\left[\mathrm{4.4.4}\right]=\frac{16}{3}$
But the line $x=\alpha$ divide above area into two equal parts.
$\therefore \frac{1}{4}{\int }_0^{\alpha }{x}^{2}dx=\frac{16}{3}\times \frac{1}{2}=\frac{8}{3}$
$\Rightarrow \frac{1}{4}\times \frac{1}{3}{\left[x^3\right]}_0^{\alpha }=\frac{8}{3}$
$\Rightarrow {\alpha }^3=32\Rightarrow {\alpha }^3=2^5\Rightarrow \alpha =2^{\frac{5}{3}}$