The area we need looks like –

We need to find the points of intersection to set the bounds of integration.

At points of intersection, y = x² + 4 = 2x²

or, x² – 4 = 0

⇒ (x + 2)(x – 2) = 0

⇒ x = -2, 2

⇒ The points of intersection are (-2, 8) and (2, 8), which is also evident from the graph.

So, the area A of the shaded region is –

A =

=

Since 4 – x² is an even function,

A =

=

=

= 32/3 (Ans)