Using the disk method, the volume by taking the region enclosed by y = ln x, x = 1 and y = 1 and rotating it around the line y = −1 is?

Let y1 = ln x , y2 = 1
When y1 = y2 ,
ln x = 1
x = e
So y1 and y2 intersect at x = e

distance( y = 1 , y = - 1 )
= 1 - (-1)
= 2

distance( y = ln x , y = - 1 )
= ( ln x ) - (-1)
= 1 + ln x

dv
= [ π*2^2 - π*( 1 + ln x )^2 ] dx
= π*[ 4 - ( 1 + ln x )^2 ] dx
= π*[ 3 - 2*ln x - ( ln x )^2 ] dx

V
= ∫ dV
= π*[ 3 - 2*ln x - ( ln x )^2 ] dx , from x = 1 to x = e