Question

Consider two curves $C_{1} : y^{2}=4[\sqrt{y}]x$ and $C_{2} : x^{2}=4 [\sqrt{x}]y$, where [.] denotes the greatest integer function. Then the area of region enclosed by these two curves within the square formed by the lines $x=1, y=1, x=4, y=4$ is

• A. $8/3$ sq. units
• B. $10/3$ sq. units
• C. $11/3$ sq. units
• D. $11/4$ sq. units

Answer 1

Answer: (C)

$y^{2}=4 \left[\sqrt{y}\right]x $
For $y \in [1,4), \left[\sqrt{y}\right]=1 $ or $y^{2}=4x$
Similarly, for $x\,\in [1,4) , ⌊\sqrt{x} ⌋=1$ and $x^{2}=4 ⌊\sqrt{x}⌋ y$ would transform into $x^{2}=4y$

The required area is the shaded region
$A=\int\limits_{1}^{2}\left(2\sqrt{x}-1\right)dx+\int\limits_{2}^{4} \left(2\sqrt{x}-\frac{x^{2}}{4}\right)dx$
$=\left(\frac{4}{3}x^{3 /2}-x\right)_{1}^{2}+\left(\frac{4}{3}x^{3/ 2}-\frac{x^{3}}{12}\right)_{2}^{4}$
$=\frac{11}{3}$ sq. units