Find the area of the region {(x, y) : y^{2}≤8x, x^{2} + y^{2}≤ 9}

To find area {(x, y): y^{2}≤ 8x, x^{2} + y^{2}≤9}

y^{2} = 8x ...(i)

x^{2} + y^{2} = 9 ...(ii)

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

Or, x^{2} + 8x = 9

Or, x^{2} + 8x – 9 = 0

Or, (x + 9)(x – 1) = 0

Or, x = – 9 or x = 1

And when x = 1 then y = ±2√2

Equation (i) represents a parabola with vertex (0,0) and axis as x – axis, equation (ii) represents a circle with centre (0,0) and radius 3 units, so it meets area at (±3, 0), (0,±3).

Point of intersection of parabola and circle is (1,2√2) and (1, – 2√2).

The sketch of the curves is as below:

Or, required area = 2(region ODCO + region DBCD)

Hence, the required area is sq. units.

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