The area of the 1st parallelogram is calculated as, Area=|a×
^
i
| The value of a×
^
i
is expressed as, a×
^
i
=(sin2x
^
i
+cos2x
^
j
+
^
k
)×
^
i
=−cos2x
^
k
+
^
j
Then, |a×
^
i
|=√cos4x+1 The area of the 2nd parallelogram is calculated as, Area=|a×
^
j
| The value of a×
^
j
is expressed as, a×
^
j
=(sin2x
^
i
+cos2x
^
j
+
^
k
)×
^
j
=sin2x
^
k
−
^
i
Then, |a×
^
j
|=√sin4x+1 The area of the 3rd parallelogram is calculated as, Area=|a×
^
k
| The value of a×
^
k
is expressed as, a×
^
k
=(sin2x
^
i
+cos2x
^
j
+
^
k
)×
^
k
=−sin2x
^
j
+cos2x
^
i
Then, |a×
^
k
|=√sin4x+cos4x The sum of the area is calculated as, A=cos4x+1+sin4x+1+sin4x+cos4x =4−(sin2x)2 since, −1≤sin2x≤1 0≤(sin2x)2≤1 3≤A≤4 Thus, A lies in the interval [3,4].