Find the area of the region {(x, y) : y2≤8x, x2 + y2≤ 9}

To find area {(x, y): y2≤ 8x, x2 + y2≤9}

y2 = 8x ...(i)


x2 + y2 = 9 ...(ii)


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


Or, x2 + 8x = 9


Or, x2 + 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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