The area bounded by the curves `y=(log)_e xa n dy=((log)_e x)^2` is

Options

A. `e-2` sq. units
B. `3-e` sq. units
C. `e` sq. units
D. `e-1` sq. units

Answer

B

Explanation

Given curves are `y=log_(e) x and y = (log_(e)x )^(2)`
Solving `log_(e) x= (log_(e)x)^(2)rArrlog_(e) x=0, 1 rArr x=1 and x=e`
Also, for `1ltxlte,0ltlog_(e) x lt 1 rArr log_(e) xgt (log_(e)x)^(2)`
`"For "xgte, log_(e) x lt (log_(e)x)^(2)`
`y=(log_(e)x)^(2)gt0" for all "xgt0`
`"and when "xrarr0,(log_(e)x)^(2)rarroo`,
From these information, we can plot the graph of the functions.

`therefore" Required area "=overset(e)underset(1)int(log x-(log_(e)x)^(2))dx`
`=overset(e)underset(1)intlog xdx -overset(e)underset(1)int(log_(e)x)^(2)dx`
`=[x log_(e)x-x]_(1)^(e)-[x (log_(e)x)^(2)]_(1)^(e)+overset(e)underset(1)int(2log_(e)x)/(x)xdx`
`=1-e+2[xlog_(e)x-x]_(1)^(e)`
`=3-e` sq. units