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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 x2 + 4 – 4x = 4x + 4
Or x2 – 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