Discussion Forum



1)

If the line $x=\alpha$ divides the area of region R={(x,y) $\in$ R2 : x3≤ y≤ x, o≤x≤1 } into two equal parts, then


A) $2\alpha^{4}-4\alpha^{2}+1=0$

B) $\alpha^{4}+4\alpha^{2}-1=0$

C) $\frac{1}{2}<\alpha<1$

D) $0<\alpha\leq\frac{1}{2}$

Answer:

Option A,C

Explanation:

$\int_{0}^{1} (x-x^{3})dx=2\int_{0}^{\alpha} (x-x^{3})dx$

                    $\frac{1}{4}=2(\frac{\alpha^{2}}{2}-\frac{\alpha^{4}}{4})$

$2\alpha^{4}-4\alpha^{2}+1=0$

                       $\alpha^{2}=\frac{4-\sqrt{16-8}}{4}(\because\alpha\in(0,1))$

                  $\alpha^{2}=1-\frac{1}{\sqrt{2}}$