Find the area of the region bounded by the curves y = x – 1 and (y – 1)^{2} = 4 (x + 1).

To find region bounded by curves

y = x – 1 ...(i)

(y – 1)^{2} = 4 (x + 1) ...(ii)

On solving the equation (i) and (ii),

Or (x – 1 – 1)^{2} = 4 (x + 1)

Or (x – 2)^{2} = 4 (x + 1)

Or x^{2} + 4 – 4x = 4x + 4

Or x^{2} – 8x = 0

Or x = 0 or 8

∴y = – 1 or7

Equation (i) represents a line passing through (1,0) and (0, – 1)

Equation (ii) represents a parabola with vertex ( – 1,1) passes through (0,3),(0, – 1),.

Their points of intersection A(0, – 1) and B(8,7).

These are shown in the graph below: –

It slides from y = – 1 to y = 7,

So, required area = Region ABCDA

The area of the region bounded by the curves y = x – 1 and (y – 1)^{2} = 4 (x + 1) is

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