Draw a rough sketch of the region {(x, y) : y^{2} ≤ 3x, 3x^{2} + 3y^{2} ≤ 16} and find the area enclosed by the region using method of integration.

To find area of region{(x, y) : y^{2} ≤ 3x, 3x^{2} + 3y^{2} ≤ 16}

The given equations are,

y^{2} = 3x ...(i)

And 3x^{2} + 3y^{2} = 16

...(ii)

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 4/√3 and meet at A and B A rough sketch of the region is drawn below

Required area = Region OCBAO

= 2(area of Region)

= 2(area of Region OBAO)

= 2(area of Region ODAO + area of Region DBAD)

The area enclosed by the region is

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