Sol. We have,

f = f(x) –f(y) for all x, y ∈ R, y ≠ 0... (i)

and, = 3 ... (ii)

Now, f ′ (x) =

=

=

=

⇒ f(x) = 3 log x + log C ... (iii)

[On integrating]

Putting x = y = 1 in (i), we get: f(1) = 0

Putting x = 1 in (iii), we get

f(1) = 3 log 1 + log C

⇒ log C = 0

Putting log C = 0 in (iii), we obtain

f(x) = 3 log x for all x ∈ R –{0}

In fig., we have to find the area of the shaded region. Let us slice this region into horizontal strips. For the approximating rectangle shown in fig., we have

Length = x, Width = Δ y and it can move from y = – ∞ to y = 3. So,

Required area =

= [  y = 3 log x ⇒ x = e y/3 ]

=

= 3e – 0 = 3 e sq. units.