Answer :

Given the boundaries of the area O befound are,

• Curve is y^{2} = 2y – x

• Y-axis.

Consider the curve, y^{2} = 2y –x

y^{2} – 2y = -x

by adding 1 on both sides

y^{2} – 2y + 1 = -(x-1)

(y-1)^{2} = -(x-1)

From the above equation, we can say that, the given equation is that of a parabola with vertex at A(1,1)

Consider the line x = 0 which is the y-axis, now substituting x = 0 in the curve equation we get

y^{2}- 2y = 0

y(y-2)=0

y = 0 (or)y = 2

So , the parabola meets the y-axis at 2 points, B (0,2) and •(0,0)

As per the given boundaries,

• The parabola y^{2} = 2y-x, with vertex at A(1,1).

• X= 0 which is the y-axis.

The boundaries of the region to be found are,

•Point A, where the curve y^{2} = 2y-x has the extreme end the vertex i.e. A (1,1)

•Point O, which is the origin

•Point B, where the curve y^{2} = 2y-x and the y – axis meet i.e. B (0,2)

Consider the curve,

y^{2} = 2y - x

x = 2y - y^{2}

Area of the required region = Area of OBAO.

[Using the formula ]

The Area of the required region

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