Question 935322
The lateral area is just the area of the curved sides of the cylinder.
It does not include the bases.
It is like the area of the label on a can of soup.
It is calculated by multiplying the circumference of the base times the height of the right cylinder:
{{{lateral}}}{{{area=circumference *height=(2*pi*radius)*height}}} .
The formula in your book may be something like
{{{A[L]=2pi*r*h}}} .
 
As the radius is doubled, the circumference of the base doubles,
but the height has not been changed,
so the lateral area is just doubled.
If cylinder number 1 has {{{r=x}}} , then {{{A[L1]=2pi*r*h}}} is its lateral area.
If you double the radius to make cylinder number 2, with {{{r=2x}}} ,
that cylinder has a lateral area of
{{{A[L2]=2pi*(2x)*h}}} --> {{{A[L2]=2(2pi*x*h)}}} --> {{{A[L2]=2A[L1]}}} .
 
Each base (top and bottom) is a circle of area
{{{A[B]=pi*r^2}}}
The total surface area of the cylinder is the lateral area plus the area of the bases: 
{{{A=A[L]+2A[B]}}} or {{{A=2pi*r*h+2pi*r^2}}} .
 
As the radius is doubled, the area of each base quadruples,
making the total area more than double.
If cylinder number 1 has {{{r=x}}} , then {{{A[1]=2pi*r*h+2pir^2}}} is its total surface area.
If you double the radius to make cylinder number 2, with {{{r=2x}}} ,
that cylinder has a surface area of
{{{A[2]=2pi*(2x)*h+2pi*(2x)^2}}} --> {{{A[2]=2(2pi*x*h)+2pi*4x^2}}} --> {{{A[2]=2(2pi*x*h)+4(2pi*x^2)}}} --> {{{A[2]=2(2pi*x*h)+2(2pi*x^2)+2(2pi*x^2)}}} --> {{{A[2]=2(2pi*x*h+2pi*x^2)+4pi*x^2)}}} --> {{{A[2]=2A[1]+4pi*x^2)}}}.