Question

Consider a region $R=\left\{(x,y)\in R^2:x^2\le y\le 2x\right\}$ If a line $y=\alpha$ divides the area of region $R$ into two equal parts, then which of the following is true?

• A. ${\alpha }^3-6{\alpha }^2+16=0$
• B. $3{\alpha }^2-8\alpha +8=0$
• C. ${\alpha }^3-6{\alpha }^{3/2}-16=0$
• D. $3{\alpha }^2-8{\alpha }^{3/2}+8=0$

Answer 1

Answer: (D)

* $y\ge x^2\Rightarrow$ upper region of
$y=x^2$ $y\le 2x\Rightarrow$ lower region of $y=2x$
According to ques, area of $OABC=2$ area of $OAC$
$\Rightarrow \underset{0}{\overset{4}{\int }}\left(\sqrt{y}-\frac{y}{2}\right)dy=2\underset{0}{\overset{a}{\int }}\left(\sqrt{y}-\frac{y}{2}\right)\cdot dy$
$\Rightarrow \frac{4}{3}=2\left[\frac{2}{3}{\alpha }^{3/2}-\frac{1}{4}.{\alpha }^2\right]$
$3{\alpha }^2-8{\alpha }^{3/2}+8=0$