The area of the region `{(x,y):xyle 8,le yle x^2}` is

Options

A. `8log_e2 -(14)/(3)`
B. `8 log_e 2-(7)/(3)`
C. `16 log_e2-(14)/(3)`
D. `16 log_e 2-6`

Answer

C

Explanation

The given region `{(x,y):xyle 8,1le ylex^2}`.
From the figure, region A and B satisfy the given region, but only A is bounded region

`A=int _(1)^(2)(x62-1)dx+int_(2)^(8)((8)/(x)-1)dx " "[because "Points" P(1,1),Q(2,4) and R(8,1)]`
`=[(x^3)/(3)-x]^2+[8log |x|-x_2]_(2)^(8)`
`=((8)/(3)-2(1)/(3)+1)+8log 8-8log 2+2`
`=-(14)/(3)+16 log 2=16 log 2-(14)/(3)`.