y = x⁴ – 2x³ + x² + 3

⇒ y’ = 4x³ – 6x² + 2x

At extrema of y, y’ = 0

i.e., 4x³ – 6x² + 2x = 0

or, 2x³ – 3x² + x = 0

⇒ x(2x – 1)(x – 1) = 0

i.e., x = 0, x = 1/2 and x = 1 correspond to extrema of y

Now, y’’ = 12x² – 12x + 2

y’’₀ = 2 > 0 ⇒ x = 0 corresponds to a minima

y’’_1/2 = -1 < 0 ⇒ x = 1/2 corresponds to a maxima

y’’₁ = 2 > 0 ⇒ x = 1 corresponds to a minima

So, x = 0 and x = 1 are the ordinates we need.

So, the area A is –

A =

=

(Ans)