Consider two curves `C_1: y^2=4[sqrt(y)]x a n dC_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 `8/3s qdotu n i t s` (b) `(10)/3s qdotu n i t s` `(11)/3s qdotu n i t s` (d) `(11)/4s qdotu n i t s`

Options

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

Answer

C

Explanation

`y^(2)=4[sqrt(y)]x`
`"For "y in [1,4),[sqrt(y)]=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=overset(2)underset(0)int(2sqrt(x)-1)dx+overset(4)underset(2)int(2sqrt(x)-(x^(2))/(4))dx`
`=((4)/(3)x^(3//2)-x)_(1)^(2)+((4)/(3)x^(3//2)-(x^(3))/(12))_(2)^(4)=(11)/(3)` sq. units